#501845

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	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

679 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)(13,14,17,20)(15,24,25,38)(16,19,31,34)(18,32,30,23)(21,37,36,26)(22,40,35,28)(27,44,33,41)(29,42,43,39);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(14,19)(15,25)(16,28)(18,30)(20,34)(21,38)(22,23)(24,36)(26,43)(29,37)(31,40)(32,35)(39,41)(42,44);(2,4)(6,10)(9,12)(14,20)(19,34)(21,36)(22,35)(23,32)(24,38)(39,42)(41,44);(13,15,17,25)(14,21,20,36)(16,29,31,43)(18,33,30,27)(19,24,34,38)(22,41,35,44)(23,42,32,39)(26,40,37,28)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,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, 409041);