#7273453

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
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

1462 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)(17,18,24,29,23,28,22,27)(19,30,26,20,31,25,21,32);(2,5)(3,7)(9,10)(11,16)(12,14)(13,15)(17,19,23,31)(18,25,28,30)(20,29,32,27)(21,24,26,22);(9,11,12,15)(10,13,14,16)(17,20,23,32)(18,26,28,21)(19,27,31,29)(22,25,24,30);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,21,23,26)(18,20,28,32)(19,22,31,24)(25,27,30,29);(2,5)(6,8)(10,14)(13,16)(17,22,23,24)(18,27,28,29)(19,26,31,21)(20,25,32,30)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,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, 7180649);