#7090236

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

1708 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)(17,18,24,27)(19,28,31,32)(20,25,29,30)(21,22,26,23);(9,10)(11,15)(12,14)(13,16)(17,19,22,25,24,31,23,30)(18,20,21,32,27,29,26,28);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,24,29)(18,25,27,30)(19,21,31,26)(22,32,23,28);(11,16)(13,15)(17,21)(18,22)(19,29)(20,31)(23,27)(24,26)(25,32)(28,30);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,22,24,23)(18,26,27,21)(19,25,31,30)(20,28,29,32)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=ghi,e^2=gi,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, 6997432);