#509315

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

584 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)(7,8)(9,11)(10,12)(13,14,17,23)(15,25,19,20)(16,28,30,38)(18,21,32,37)(22,31,40,42)(24,27,35,33)(26,41,29,43)(34,36,44,39);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,17,19)(14,20,23,25)(16,29,30,26)(18,27,32,33)(21,24,37,35)(22,39,40,36)(28,41,38,43)(31,44,42,34);(2,4)(5,8,9,10)(6,7,11,12)(14,21)(15,19)(18,33)(20,35)(22,38)(23,37)(24,25)(26,29)(27,32)(28,40)(31,34)(36,43)(39,41)(42,44);(6,11)(7,12)(13,16,17,30)(14,22,23,40)(15,26,19,29)(18,34,32,44)(20,36,25,39)(21,38,37,28)(24,41,35,43)(27,42,33,31)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ei,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=de,f^c=fh,h^c=hi,f^d=fh,h^d=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, 416511);