#6576105

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573643
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	C₂²×C₄	C₂⁵
prime radical	#24461	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1860 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483301);