#434689

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×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1158 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)(9,10,13,18,15,20,14,19)(11,21,24,32,25,33,23,16)(12,26,28,34,30,35,29,17)(22,37,39,40,27,31,38,36);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,27)(13,23)(14,24)(15,25)(17,31)(18,33)(19,21)(20,32)(22,30)(26,40)(28,39)(29,38)(34,37)(35,36);(1,3,4,7)(2,6,5,8)(9,12)(10,17,20,34)(11,22)(13,28)(14,29)(15,30)(16,31,32,37)(18,26,19,35)(21,36,33,40)(23,38)(24,39)(25,27);(2,5)(6,8)(10,18)(11,23)(13,14)(16,32)(17,26)(19,20)(22,38)(24,25)(27,39)(28,29)(31,37)(34,35)

Polycyclic presentation:

2^9|a^2=fg,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, 341885);