#7074651

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

1590 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fhi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6981847);