#6352947

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352941
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

2955 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260143);