#428801

order	512
nilpotency class	3
isoclinism	isoclinic class of #418518
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1027 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ef,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335997);