#7337950

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324468
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 ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1654 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=fi,e^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=dh,e^c=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, 7245146);