#6909014

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1551 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=gi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6816210);