#7330576

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1595 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fgi,d^2=f,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^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7237772);