#6793792

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	#24378	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1675 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,6)(5,8)(17,18,22,23,21,27,19,26)(20,30,28,32,31,25,29,24);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,13)(14,16)(17,19,21,22)(18,23,27,26)(20,29,31,28)(24,30,32,25);(10,13)(14,15)(17,20)(18,24)(19,28)(21,31)(22,29)(23,25)(26,30)(27,32);(2,6)(5,7)(17,20)(18,25)(19,29)(21,31)(22,28)(23,24)(26,32)(27,30);(3,8)(5,7)(9,11,12,16)(10,14,13,15)(17,19,21,22)(18,26,27,23)(20,28,31,29)(24,30,32,25)

Polycyclic presentation:

2^9|a^2=g,b^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,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, 6700988);