#7090227

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

1815 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)(19,27,32,28)(20,21,25,22)(24,30,29,31);(9,10)(11,15)(12,14)(13,16)(17,19,22,31,23,32,21,30)(18,24,25,27,26,29,20,28);(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)(24,31)(27,32)(29,30);(11,16)(13,15)(17,20)(18,21)(19,29)(22,26)(23,25)(24,32)(27,31)(28,30);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,23,22)(18,25,26,20)(19,30,32,31)(24,27,29,28)

Polycyclic presentation:

2^9|a^2=i,b^2=f,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=df,e^b=eh,d^c=dh,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, 6997423);