#6352967

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

2977 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:

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

Polycyclic presentation:

2^9|c^2=hi,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, 6260163);