#501847

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
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×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

696 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)(15,25,19,21)(16,28,29,39)(17,20,33,35)(22,38,40,30)(23,27,36,31)(26,43,34,41)(32,42,44,37);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(14,20)(15,19)(16,29)(17,31)(21,36)(22,39)(23,25)(24,35)(27,33)(28,40)(30,44)(32,38)(37,41)(42,43);(2,4)(6,10)(9,12)(13,15,18,19)(14,21,24,25)(16,26,29,34)(17,27,33,31)(20,23,35,36)(22,37,40,42)(28,41,39,43)(30,32,38,44);(13,16,18,29)(14,22,24,40)(15,26,19,34)(17,32,33,44)(20,28,35,39)(21,37,25,42)(23,41,36,43)(27,38,31,30)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,e^b=eh,h^b=hi,e^d=eh,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409043);