#6793899

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×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1502 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)(17,18,23,25,21,27,22,28)(19,29,37,39,30,24,32,41)(20,33,38,44,35,26,36,42)(31,40,46,48,34,43,45,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,13)(14,16)(17,19)(18,24)(20,34)(21,30)(22,32)(23,37)(25,41)(26,43)(27,29)(28,39)(31,35)(33,40)(36,46)(38,45)(42,47)(44,48);(10,13)(14,15)(18,25)(19,30)(22,23)(24,39)(26,42)(27,28)(29,41)(31,34)(33,44)(36,38)(40,47)(43,48);(2,6)(5,7)(17,20,21,35)(18,26,27,33)(19,31,30,34)(22,36,23,38)(24,40,29,43)(25,42,28,44)(32,45,37,46)(39,47,41,48);(3,8)(5,7)(9,11,12,16)(10,14,13,15)(20,35)(26,33)(31,34)(36,38)(40,43)(42,44)(45,46)(47,48)

Polycyclic presentation:

2^9|a^2=gi,d^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, 6701095);