#6796183

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1270 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)(3,5)(4,7)(6,8)(17,18,23,30,22,26,21,25)(19,24,29,32,20,27,28,31);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,13)(14,16)(17,19,22,20)(18,24,26,27)(21,28,23,29)(25,31,30,32);(2,3)(6,7)(10,13)(14,15)(17,18,22,26)(19,27,20,24)(21,30,23,25)(28,31,29,32);(1,4,5,8)(2,6,3,7)(17,20,22,19)(18,24,26,27)(21,29,23,28)(25,31,30,32);(9,11,12,16)(10,14,13,15)(19,20)(24,27)(28,29)(31,32)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=i,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6703379);