#6360767

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

1827 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)(11,21,25,36)(12,17,27,34)(13,19,14,18)(16,29,32,24)(22,37,40,31)(23,38,39,30)(26,35,28,33);(2,6)(5,8)(9,11,15,25)(10,16,20,32)(12,22,27,40)(13,24,14,29)(17,30,34,38)(18,21,19,36)(23,28,39,26)(31,35,37,33);(1,2)(3,8)(4,6)(5,7)(9,12)(10,17)(11,22)(13,28)(14,26)(15,27)(16,30)(18,35)(19,33)(20,34)(21,37)(23,29)(24,39)(25,40)(31,36)(32,38);(9,12)(10,17)(11,23)(13,26)(14,28)(15,27)(16,31)(18,33)(19,35)(20,34)(21,38)(22,29)(24,40)(25,39)(30,36)(32,37);(9,13,15,14)(10,18,20,19)(11,24,25,29)(12,26,27,28)(16,21,32,36)(17,33,34,35)(22,23,40,39)(30,31,38,37)

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,f^b=fh,e^c=eh,g^c=gi,f^d=fh

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, 6267963);