#506182

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
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

718 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)(13,14)(15,23)(16,20)(17,21)(18,22)(19,25)(24,32)(26,39)(27,38)(28,35)(29,43)(30,37)(31,34)(33,40)(36,41)(42,44);(5,7,8,12)(6,9,10,11)(13,15,24,40,28,31,30,29,16,25,17,26,18,27,42,41)(14,19,32,39,35,38,37,36,20,23,21,33,22,34,44,43);(2,4)(13,15,16,25)(14,19,20,23)(17,29,24,41)(18,31,28,27)(21,36,32,43)(22,38,35,34)(26,42,40,30)(33,44,39,37);(6,10)(9,11)(14,20)(19,23)(21,32)(22,35)(33,39)(34,38)(36,43)(37,44)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413378);