#6352900

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352701
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×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2943 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:

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

Polycyclic presentation:

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

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, 6260096);