#6360769

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

Polycyclic presentation:

2^9|a^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,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, 6267965);