#6690467

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1864 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,18,23,24)(19,27,31,25)(20,22,26,21)(28,30,29,32);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,19,23,31)(18,25,24,27)(20,30,26,32)(21,29,22,28);(9,12,13,11)(10,14,15,16)(17,20)(18,22)(19,28)(21,24)(23,26)(25,32)(27,30)(29,31);(2,6)(5,7)(17,21)(18,26)(19,29)(20,24)(22,23)(25,32)(27,30)(28,31);(3,8)(5,7)(17,22,23,21)(18,26,24,20)(19,30,31,32)(25,28,27,29)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285106
maximal normal subgroup of	#72285106

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6597663);