#7273304

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234437
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	#42752	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1457 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)(3,6)(4,5)(7,8)(11,15)(13,16)(17,18,24,29,23,27,22,28)(19,25,21,26,30,32,31,20);(2,5)(3,7)(9,10)(11,16)(12,14)(13,15)(17,19)(18,25)(20,28)(21,24)(22,31)(23,30)(26,29)(27,32);(9,11,12,15)(10,13,14,16)(17,20)(18,21)(19,29)(22,25)(23,26)(24,32)(27,31)(28,30);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,21)(18,26)(19,22)(20,27)(23,31)(24,30)(25,28)(29,32);(2,5)(6,8)(10,14)(13,16)(18,27)(19,30)(21,31)(28,29)

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,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, 7180500);