#6544053

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533141
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3241 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,d^2=hi,e^2=h,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,g^b=gi,f^c=fi,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, 6451249);