#7090135

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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1523 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,20)(19,25,27,32)(21,29,31,24)(23,30,28,26);(9,10)(11,15)(12,14)(13,16)(17,19,22,27)(18,23,20,28)(21,30,31,26)(24,32,29,25);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,22,18)(19,23,27,28)(21,29,31,24)(25,30,32,26);(11,16)(13,15)(19,26)(21,31)(23,25)(24,29)(27,30)(28,32);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,20)(23,28)(24,29)(25,32)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,f^b=fi,d^c=dhi,e^c=ei,f^d=fi

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, 6997331);