#509999

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
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	#24778	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

600 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)(5,6,9,12)(7,8,11,10)(13,14,17,20)(15,24,27,21)(16,28,19,22)(18,23,30,34)(25,35,42,37)(26,36,33,38)(29,40,31,39)(32,44,43,41);(5,7,9,11)(6,10,12,8)(14,20)(15,25)(16,19)(18,31)(21,35)(23,40)(24,37)(26,43)(27,42)(29,30)(32,33)(34,39)(36,44)(38,41);(2,4)(5,8,9,10)(6,11,12,7)(13,15,17,27)(14,21,20,24)(16,26,19,33)(18,32,30,43)(22,36,28,38)(23,41,34,44)(25,29,42,31)(35,39,37,40);(2,4)(6,12)(7,11)(13,16,17,19)(14,22,20,28)(15,26,27,33)(18,29,30,31)(21,36,24,38)(23,39,34,40)(25,32,42,43)(35,41,37,44)

Polycyclic presentation:

2^9|a^2=efi,b^2=ei,c^2=e,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gh,h^b=hi,g^c=gh,h^c=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, 417195);