#7509384

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1335 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)(3,4)(5,7)(6,8)(9,10)(11,14)(12,16)(13,15)(19,26)(22,31)(23,32)(27,30);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16)(17,18,20,24)(19,23,26,32)(21,29,28,25)(22,30,31,27);(9,12,13,11)(10,14,15,16)(17,19)(18,22)(20,26)(21,30)(23,29)(24,31)(25,32)(27,28);(2,7)(3,8)(10,15)(14,16)(17,19)(18,23)(20,26)(21,27)(22,29)(24,32)(25,31)(28,30);(1,4,5,6)(2,3,7,8)(19,26)(22,31)(23,32)(27,30)

Polycyclic presentation:

2^9|b^2=g,c^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7416580);