#72341231

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#32898
commutator, Frattini, cosocle	C₂²×C₂²⋊C₄	C₂⁴
prime radical	#8926669	C₂
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#120 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

4010 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^10|b^2=hj,d^2=j,g^2=hj,b^a=be,c^a=cij,c^b=cef,d^a=dhj,d^b=dfghj,d^c=di,e^c=ej,e^d=eh,f^c=fj,f^d=fhi,g^a=gh,g^b=gh,g^c=gij,g^d=gi,g^f=gj,i^b=ij

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341243
maximal normal subgroup of	#72341243

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, 2, F.8*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 7, F.8*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);