#509497

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

667 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=e,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=dei,f^c=fh,h^c=hi,f^d=fh,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, 416693);