#6612222

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1714 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18,23,27)(19,28,31,32)(20,25,24,30)(21,26,22,29);(9,11)(10,13)(12,16)(14,15)(17,19,21,30,23,31,22,25)(18,24,29,32,27,20,26,28);(2,5)(6,8)(11,16)(13,14)(17,20)(18,25)(19,29)(21,28)(22,32)(23,24)(26,31)(27,30);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,16)(13,15)(17,21,23,22)(18,26,27,29)(19,30,31,25)(20,32,24,28);(11,16)(13,14)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,d^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,d^c=dh,e^d=ehi

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, 6519418);