#7036740

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994107
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	#2537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1402 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:

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

Polycyclic presentation:

2^9|a^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6943936);