#6566264

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535696
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1669 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6473460);