#428802

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

Polycyclic presentation:

2^9|a^2=ef,d^2=efi,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, 335998);