#6352553

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

2933 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:

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

Polycyclic presentation:

2^9|a^2=hi,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, 6259749);