#6581598

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1372 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fgi,c^2=f,d^2=f,e^2=i,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, 6488794);