#424384

order	512
nilpotency class	3
isoclinism	isoclinic class of #418466
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1102 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=ef,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,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, 331580);