#504158

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, 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

547 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)(7,10,8,12)(13,14,18,24)(15,21,19,33)(16,27,28,39)(17,20,32,36)(22,38,40,29)(23,26,35,30)(25,43,34,42)(31,37,44,41);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(14,20)(15,19)(16,28)(17,30)(21,23)(22,39)(24,36)(26,32)(27,40)(29,44)(31,38)(33,35)(37,43)(41,42);(2,4)(5,8,9,7)(6,10,11,12)(13,15,18,19)(14,21,24,33)(16,25,28,34)(17,26,32,30)(20,35,36,23)(22,37,40,41)(27,43,39,42)(29,31,38,44);(13,16,18,28)(14,22,24,40)(15,25,19,34)(17,31,32,44)(20,27,36,39)(21,37,33,41)(23,42,35,43)(26,38,30,29)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^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, 411354);