#7090491

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

1820 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,26)(19,27,31,28)(20,21,25,22)(24,29,32,30);(9,10)(11,15)(12,14)(13,16)(17,19,21,29,23,31,22,30)(18,24,20,28,26,32,25,27);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20)(18,22)(19,28)(21,26)(23,25)(24,30)(27,31)(29,32);(11,16)(13,15)(17,20)(18,21)(19,24)(22,26)(23,25)(27,29)(28,30)(31,32);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,23,22)(18,25,26,20)(19,29,31,30)(24,27,32,28)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=ghi,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=eh,d^c=dh,e^c=ei,f^d=fi

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, 6997687);