#6586025

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈×#120
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×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1724 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,15,16,13)(17,18,21,24)(19,23,20,25)(22,26,29,32)(27,31,28,30);(1,3,5,4)(2,6,7,8)(17,19,21,20)(18,23,24,25)(22,27,29,28)(26,31,32,30);(9,11,12,16)(10,13,14,15)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31);(1,4,5,3)(2,6,7,8)(17,20,21,19)(18,25,24,23)(22,28,29,27)(26,30,32,31);(19,27)(20,28)(22,29)(23,31)(25,30)(26,32)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=gi,d^2=fi,h^2=i,b^a=bf,c^a=cg,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

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6493221);