#6359136

order	512
nilpotency class	3
isoclinism	isoclinic class of #6359112
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1717 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)(3,6)(4,7)(5,9)(8,12)(10,13)(11,15)(14,16)(17,18,23,28)(19,29,31,32)(20,25,24,30)(21,27,22,26);(1,3,5,11)(2,6,9,15)(4,10,12,16)(7,13,8,14)(17,19)(18,24)(20,28)(21,30)(22,25)(23,31)(26,32)(27,29);(2,7)(4,12)(6,13)(8,9)(10,16)(14,15)(17,20)(18,19)(21,32)(22,29)(23,24)(25,26)(27,30)(28,31);(17,20)(18,25)(19,26)(21,29)(22,32)(23,24)(27,31)(28,30);(17,21,23,22)(18,26,28,27)(19,30,31,25)(20,29,24,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6266332);