#7336645

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324820
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1586 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)(3,6,4,8)(17,18)(19,26)(20,24)(21,23)(22,30)(25,29)(27,32)(28,31);(1,2)(3,8)(4,6)(5,7)(9,10)(11,15)(12,14)(13,16)(18,23)(19,27)(24,31)(25,30);(9,11)(10,13)(12,16)(14,15)(17,19,22,24,21,27,29,31)(18,20,25,26,23,28,30,32);(1,3,5,4)(2,6,7,8)(10,14)(11,16)(17,20,21,28)(18,24,23,31)(19,25,27,30)(22,26,29,32);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,23)(20,28)(25,30)(26,32)

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dhi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7243841);