#6360387

order	512
nilpotency class	3
isoclinism	isoclinic class of #6360376
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

1816 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(2,5)(3,7)(6,8)(9,10,15,20)(11,21,25,31)(12,17,28,35)(13,19,14,18)(16,23,32,24)(22,37,38,33)(26,40,39,30)(27,36,29,34);(2,6)(5,8)(9,11,15,25)(10,16,20,32)(12,26,28,39)(13,24,14,23)(17,33,35,37)(18,21,19,31)(22,29,38,27)(30,36,40,34);(1,2)(3,8)(4,6)(5,7);(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,34)(19,36)(20,35)(21,37)(23,26)(24,39)(25,38)(31,33)(32,40);(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,27,28,29)(16,31,32,21)(17,34,35,36)(22,26,38,39)(30,33,40,37)

Polycyclic presentation:

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

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, 6267583);