#6355105

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355044
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1657 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262301);