#6445216

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439035
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1957 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)(3,6)(4,5)(7,8)(9,10,13,18)(11,20,23,21)(12,17,26,33)(14,28,15,19)(16,29,32,24)(22,35,38,36)(25,27,34,30)(31,40,37,39);(2,5)(6,8)(9,10)(11,21)(12,25)(13,18)(14,19)(15,28)(16,24)(17,30)(20,23)(22,37)(26,34)(27,33)(29,32)(31,38)(35,40)(36,39);(9,11,15,29,13,23,14,24)(10,16,28,21,18,32,19,20)(12,22,30,40,26,38,27,39)(17,31,34,36,33,37,25,35);(9,12)(10,17)(11,22)(13,26)(14,27)(15,30)(16,31)(18,33)(19,25)(20,35)(21,36)(23,38)(24,39)(28,34)(29,40)(32,37);(1,3,4,7)(2,6,5,8)(11,23)(16,32)(20,21)(22,38)(24,29)(31,37)(35,36)(39,40)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,h^b=hi,e^c=ei,h^c=hi,h^d=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, 6352412);