#6652433

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632088
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1193 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26,31,36)(27,32,37,47)(28,34,42,52)(29,35,30,33)(38,46,40,50)(39,48,43,45)(41,49,44,51)(53,55,54,56);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,18,24)(12,20,13,19)(16,23,17,22)(25,27)(26,32)(28,40)(29,43)(30,39)(31,37)(33,48)(34,50)(35,45)(36,47)(38,42)(41,53)(44,54)(46,52)(49,55)(51,56);(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,22,24,23)(26,33)(27,37)(29,30)(32,45)(34,49)(35,36)(38,40)(41,44)(46,55)(47,48)(50,56)(51,52);(2,5)(6,7)(9,13,14,12)(10,17,18,16)(11,20,21,19)(15,23,24,22)(25,28,31,42)(26,34,36,52)(27,38,37,40)(29,44,30,41)(32,46,47,50)(33,49,35,51)(39,54,43,53)(45,55,48,56);(1,3,4,8)(2,6,5,7)(10,18)(15,24)(16,17)(22,23)(25,29,31,30)(26,33,36,35)(27,39,37,43)(28,41,42,44)(32,45,47,48)(34,51,52,49)(38,53,40,54)(46,56,50,55)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=hi,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6559629);