#72312757

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#14716
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁴
prime radical	#6347494	C₂
LCS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

3958 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=i,b^2=h,f^2=j,b^a=bh,c^a=cf,d^a=de,d^b=dg,e^b=ei,e^c=eij,f^a=fj,f^c=fj,f^d=fij,g^a=gj,g^b=gij,h^d=hij

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312782, #72312812, #72312951
maximal normal subgroup of	#72312782, #72312812, #72312951

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