#7014865

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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

1657 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6922061);