#7055754

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#42928	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1596 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)(9,10)(11,13)(12,14)(15,16)(17,18,21,27)(19,25,29,23)(20,26,22,31)(24,30,32,28);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,29)(18,23,27,25)(20,30,22,28)(24,31,32,26);(9,11,12,16)(10,13,14,15)(17,19,21,29)(18,24,27,32)(20,28,22,30)(23,31,25,26);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,19)(18,25)(20,30)(21,29)(22,28)(23,27)(24,31)(26,32);(10,14)(13,15)(17,20,21,22)(18,26,27,31)(19,28,29,30)(23,32,25,24)

Polycyclic presentation:

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

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, 6962950);