#509721

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, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

483 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=i,d^2=e,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, 416917);