#6801565

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1551 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)(3,7)(4,6)(5,8)(17,18,23,36,22,27,21,26)(19,28,30,24,32,38,31,39)(20,25,34,41,33,40,35,42)(29,43,46,37,44,48,45,47);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(20,33)(25,40)(29,44)(34,35)(37,47)(41,42)(43,48)(45,46);(10,13)(14,15)(17,19,22,32)(18,24,27,39)(20,29,33,44)(21,30,23,31)(25,37,40,47)(26,38,36,28)(34,45,35,46)(41,43,42,48);(2,6)(5,7)(17,20)(18,25)(19,29)(21,35)(22,33)(23,34)(24,37)(26,42)(27,40)(28,43)(30,46)(31,45)(32,44)(36,41)(38,48)(39,47);(3,8)(5,7)(9,11,12,16)(10,14,13,15)(17,21,22,23)(18,26,27,36)(19,30,32,31)(20,34,33,35)(24,38,39,28)(25,41,40,42)(29,45,44,46)(37,43,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6708761);