#6352549

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×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2812 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=i,e^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, 6259745);