#7091209

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
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1272 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=i,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,f^b=fi,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, 6998405);