#506205

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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

691 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)(7,11)(8,10)(9,12)(14,19)(21,32)(22,33)(23,34)(24,41)(37,40)(38,44)(39,43);(5,7,8,12)(6,9,10,11)(13,14,20,33,28,34,30,39,16,19,17,22,18,23,36,43)(15,24,42,37,31,44,29,21,25,41,26,40,27,38,35,32);(2,4)(13,15,16,25)(14,21,19,32)(17,29,20,35)(18,31,28,27)(22,38,33,44)(23,40,34,37)(24,43,41,39)(26,36,42,30);(6,10)(9,11)(14,19)(21,32)(22,33)(23,34)(24,41)(37,40)(38,44)(39,43)

Polycyclic presentation:

2^9|b^2=eg,c^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,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, 413401);