#6445213

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

2015 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,25)(13,18)(14,19)(16,29)(17,27)(20,37)(21,23)(22,35)(24,32)(26,31)(28,36)(30,34)(33,38)(39,40);(2,5)(6,8)(9,11)(10,15)(12,26)(13,22)(14,24)(16,33)(17,21)(18,35)(19,32)(20,30)(23,27)(25,31)(28,39)(29,38)(34,37)(36,40);(9,12,27,30,14,28,13,29)(10,16,18,36,19,34,17,25)(11,20,23,39,24,38,22,26)(15,31,35,33,32,40,21,37);(9,10)(11,21)(12,16)(13,17)(14,19)(15,23)(18,27)(20,37)(22,32)(24,35)(25,29)(26,40)(28,34)(30,36)(31,39)(33,38);(1,3,4,7)(2,6,5,8)(12,28)(16,34)(20,38)(25,36)(26,39)(29,30)(31,40)(33,37)

Polycyclic presentation:

2^9|c^2=gi,e^2=f,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, 6352409);