#6445192

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

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

Polycyclic presentation:

2^9|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=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, 6352388);