#425443

order	512
nilpotency class	3
isoclinism	isoclinic class of #418484
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

997 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=ef,d^2=e,e^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^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, 332639);