#442234

order	512
nilpotency class	3
isoclinism	isoclinic class of #421903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

823 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ei,d^2=f,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, 349430);