#7155636

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996671
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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1587 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fgi,d^2=i,e^2=h,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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7062832);