#423948

order	512
nilpotency class	3
isoclinism	isoclinic class of #418492
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
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

1233 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)(6,7)(9,10)(11,18)(12,26)(13,28)(14,20)(15,22)(16,21)(17,24)(19,31)(23,38)(25,34)(27,33)(29,40)(30,37)(32,36)(35,39);(1,2,4,5)(3,7,8,6)(9,11)(10,17)(12,16)(13,29)(14,24)(15,25)(18,20)(19,35)(21,34)(22,26)(23,30)(27,31)(28,38)(32,39)(33,36)(37,40);(9,12,15,11,14,25,16,24)(10,18,21,26,20,17,22,34)(13,27,31,23,30,39,32,29)(19,35,28,38,36,33,37,40);(1,3,4,8)(2,6,5,7)(9,13,14,30)(10,19,20,36)(11,23,24,29)(12,27,25,39)(15,31,16,32)(17,33,18,35)(21,28,22,37)(26,38,34,40)

Polycyclic presentation:

2^9|b^2=ei,c^2=f,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,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, 331144);