#6445193

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439025
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)(9,10,14,19)(11,21,25,22)(12,17,28,33)(13,18,15,20)(16,29,32,24)(23,35,38,36)(26,30,34,27)(31,39,40,37);(2,5)(6,8)(9,10)(11,22)(12,26)(13,20)(14,19)(15,18)(16,24)(17,27)(21,25)(23,31)(28,34)(29,32)(30,33)(35,37)(36,39)(38,40);(9,11,15,29,14,25,13,24)(10,16,18,22,19,32,20,21)(12,23,30,39,28,38,27,37)(17,31,26,36,33,40,34,35);(9,12)(10,17)(11,23)(13,27)(14,28)(15,30)(16,31)(18,26)(19,33)(20,34)(21,35)(22,36)(24,37)(25,38)(29,39)(32,40);(1,3,4,7)(2,6,5,8)(9,13,14,15)(10,18,19,20)(11,24,25,29)(12,27,28,30)(16,22,32,21)(17,26,33,34)(23,37,38,39)(31,36,40,35)

Polycyclic presentation:

2^9|a^2=i,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=dh,e^b=ei,h^b=hi,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, 6352389);