#72306435

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	C₂×#3499	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

5795 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,13)(4,9)(5,15)(6,11)(7,12)(8,10)(14,16)(17,18,23,27)(19,22)(20,25,34,43)(21,35,39,26)(24,30)(28,40)(29,41)(31,33)(32,44,37,42)(36,38)(45,47)(46,48);(1,3,6,10,7,15,4,14)(2,8,9,13,12,16,11,5)(17,19,23,30)(18,24)(20,31,34,28)(21,36,39,29)(22,27)(25,40)(26,38)(32,45,37,46)(33,43)(35,41)(42,47)(44,48);(1,4,7,6)(2,9,12,11)(3,10,15,14)(5,16,13,8)(17,20,23,34)(18,25,27,43)(19,28,30,31)(21,37,39,32)(22,40,24,33)(26,44,35,42)(29,46,36,45)(38,47,41,48);(1,5,7,13)(2,10,12,14)(3,11,15,9)(4,8,6,16)(17,21,23,39)(18,26,27,35)(19,29,30,36)(20,32,34,37)(22,38,24,41)(25,42,43,44)(28,45,31,46)(33,47,40,48);(1,4)(2,11)(5,13)(6,7)(9,12)(10,14)(17,22,23,24)(18,19,27,30)(20,33,34,40)(21,38,39,41)(25,31,43,28)(26,29,35,36)(32,47,37,48)(42,46,44,45)

Polycyclic presentation:

2^10|a^2=f,b^2=fgi,c^2=ij,d^2=ij,e^2=i,g^2=j,h^2=j,b^a=bg,c^b=cij,d^a=dfhj,d^b=dfhj,d^c=dij,e^a=efghij,e^b=egj,e^c=eij,e^d=efhi,f^b=fi,f^e=fi,g^a=gij,g^b=gi,g^d=gj,g^e=gj,h^a=hj,h^b=hi,h^d=hj,h^e=hij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306523, #72306526
maximal normal subgroup of	#72306523, #72306526

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 2, F.6*F.7*F.9);;
gap> SetPower(coll, 3, F.9*F.10);;
gap> SetPower(coll, 4, F.9*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);