#7074676

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993979
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 ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1600 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)(17,18,22,26,21,23,20,25)(19,27,30,31,28,32,29,24);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(18,23)(19,28)(25,26)(29,30);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,19,21,28)(18,24,23,31)(20,30,22,29)(25,27,26,32);(1,3,5,4)(2,8,6,7)(11,16)(13,15)(17,20,21,22)(18,25,23,26)(19,29,28,30)(24,32,31,27);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(18,23)(24,31)(25,26)(27,32)

Polycyclic presentation:

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

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, 6981872);