#6729702

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1594 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=gi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6636898);