#6566268

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
prime radical	#61252	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1653 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)(19,26)(20,23)(24,31)(25,29)(28,32);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,18,23,29,21,25,20,22)(19,24,30,31,26,32,27,28);(3,9)(6,12)(10,15)(13,16)(17,19,20,27,21,26,23,30)(18,24,22,28,25,32,29,31);(1,4)(2,7)(3,10)(5,11)(6,13)(8,14)(9,15)(12,16)(18,25)(22,29)(24,32)(28,31);(4,11)(7,14)(10,15)(13,16)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,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, 6473464);