#6445222

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439029
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
prime radical	#24293	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1927 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)(11,21,23,17)(12,24,26,32)(13,18,27,34)(15,29,16,20)(22,37,38,31)(25,40,39,33)(28,36,30,35);(2,5)(6,8)(9,11)(10,17)(12,15)(13,25)(14,23)(16,26)(18,33)(19,21)(20,24)(22,30)(27,39)(28,38)(29,32)(31,35)(34,40)(36,37);(9,12,16,11,14,26,15,23)(10,17,29,32,19,21,20,24)(13,25,30,22,27,39,28,38)(18,31,36,33,34,37,35,40);(9,13)(10,18)(11,22)(12,25)(14,27)(15,28)(16,30)(17,31)(19,34)(20,35)(21,37)(23,38)(24,40)(26,39)(29,36)(32,33);(1,3,4,7)(2,6,5,8)(11,23)(12,26)(17,21)(22,38)(24,32)(25,39)(31,37)(33,40)

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,e^b=ei,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, 6352418);