#6353112

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353061
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

3036 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)(4,16)(9,25)(10,23)(11,20)(12,24)(15,21)(17,19)(22,27)(26,32)(28,31);(1,2)(3,8)(4,9)(5,20)(6,11)(7,12)(10,13)(14,22)(15,24)(16,27)(17,26)(18,25)(19,28)(21,23)(29,31)(30,32);(1,3,13,21)(2,8,10,23)(4,14,17,29)(5,7,15,6)(9,22,26,31)(11,20,12,24)(16,18,19,30)(25,28,32,27);(1,4,6,16)(2,9,11,27)(3,14,5,18)(7,19,13,17)(8,22,20,25)(10,26,12,28)(15,30,21,29)(23,31,24,32);(4,17)(9,26)(14,29)(16,19)(18,30)(22,31)(25,32)(27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260308);