#6502822

order	512
nilpotency class	3
isoclinism	isoclinic class of #6502819
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6410018);