#434690

order	512
nilpotency class	3
isoclinism	isoclinic class of #419852
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1096 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,h^c=hi,e^d=ei,f^d=fi,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, 341886);