#442375

order	512
nilpotency class	3
isoclinism	isoclinic class of #421903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

690 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=eg,c^2=ei,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,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, 349571);