#6581162

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573477
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#126 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1502 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6488358);