#6804826

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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

1207 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6712022);