#6360391

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

1811 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)(11,21,25,30)(12,17,14,19)(13,18,28,36)(16,24,32,22)(23,37,39,34)(26,33,29,35)(27,40,38,31);(2,6)(5,8)(9,11,15,25)(10,16,20,32)(12,22,14,24)(13,27,28,38)(17,30,19,21)(18,34,36,37)(23,29,39,26)(31,33,40,35);(1,2)(3,8)(4,6)(5,7)(9,12,15,14)(10,17,20,19)(11,22,25,24)(13,26,28,29)(16,30,32,21)(18,33,36,35)(23,38,39,27)(31,34,40,37);(9,13)(10,18)(11,23)(12,26)(14,29)(15,28)(16,31)(17,33)(19,35)(20,36)(21,37)(22,38)(24,27)(25,39)(30,34)(32,40);(9,12,15,14)(10,19,20,17)(11,24,25,22)(13,26,28,29)(16,30,32,21)(18,35,36,33)(23,27,39,38)(31,34,40,37)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^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, 6267587);