#6352484

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

2894 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)(18,23)(20,21)(24,31)(25,29)(26,28)(30,32);(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,18,22,25)(19,24,27,32)(20,23,21,29)(26,30,28,31);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,19,22,27)(18,24,25,32)(20,26,21,28)(23,30,29,31);(4,11)(7,14)(9,15)(13,16)(17,19)(18,24)(20,28)(21,26)(22,27)(23,31)(25,32)(29,30);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,20,22,21)(18,23,25,29)(19,26,27,28)(24,30,32,31)

Polycyclic presentation:

2^9|b^2=h,c^2=h,e^2=i,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, 6259680);