#6352502

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352469
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2885 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259698);