#7230510

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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	#52985	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1494 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:

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

Polycyclic presentation:

2^9|c^2=hi,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=efi,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7137706);