#509269

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

593 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,14,19,18)(15,20,25,33)(16,21,28,34)(17,22,32,31)(23,35,40,39)(24,29,37,42)(26,36,43,41)(27,30,38,44);(5,7)(6,10)(8,11)(9,12)(13,15,19,25)(14,20,18,33)(16,26,28,43)(17,29,32,42)(21,36,34,41)(22,37,31,24)(23,38,40,27)(30,35,44,39);(2,4)(5,8,9,10)(6,7,11,12)(13,16,19,28)(14,21,18,34)(15,23,25,40)(17,30,32,44)(20,35,33,39)(22,38,31,27)(24,41,37,36)(26,42,43,29);(2,4)(6,11)(7,12)(13,14,19,18)(15,20,25,33)(16,21,28,34)(17,22,32,31)(23,35,40,39)(24,29,37,42)(26,36,43,41)(27,30,38,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ei,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416465);