#421339

order	512
nilpotency class	3
isoclinism	isoclinic class of #418458
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1002 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=f,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,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, 328535);