#7006166

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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

1718 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,23,21,26,19,27)(20,30,28,24,31,25,29,32);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,22)(18,23,26,27)(20,29,31,28)(24,25,32,30);(9,11)(10,13)(12,16)(14,15)(17,20,21,31)(18,24,26,32)(19,28,22,29)(23,30,27,25);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20)(18,25)(19,29)(21,31)(22,28)(23,32)(24,27)(26,30);(10,14)(13,15)(17,19,21,22)(18,23,26,27)(20,28,31,29)(24,30,32,25)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,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, 6913362);