#6611728

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2075 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)(12,16)(13,15)(18,22)(20,27)(23,29)(24,28)(26,31)(30,32);(9,11,13,12)(10,14,15,16)(17,18,21,24)(19,23,25,30)(20,28,27,22)(26,32,31,29);(2,5)(6,8)(10,15)(11,12)(17,19,21,25)(18,23,24,30)(20,26,27,31)(22,29,28,32);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,15,14);(11,12)(14,16)(19,25)(23,30)(26,31)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6518924);