#7019158

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1813 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=gi,c^2=i,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6926354);