#6819590

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1291 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=gh,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6726786);