#6360887

order	512
nilpotency class	3
isoclinism	isoclinic class of #6360760
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

1583 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283494
maximal normal subgroup of	#72283494

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6268083);