#6470815

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470811
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

1978 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:

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

Polycyclic presentation:

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

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, 6378011);