#504144

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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

558 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)(15,25)(16,28)(17,20)(18,24)(19,21)(22,29)(23,31)(26,41)(27,36)(30,43)(32,40)(33,35)(34,39)(37,44)(38,42);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(14,20)(15,19)(17,31)(21,36)(22,28)(23,25)(24,35)(26,34)(27,33)(29,32)(37,41)(38,43)(39,40)(42,44);(2,4)(5,8,9,7)(6,10,11,12)(13,15,18,19)(14,21,24,25)(16,26,30,34)(17,27,33,31)(20,23,35,36)(22,37,38,40)(28,39,43,41)(29,32,42,44);(13,16,18,30)(14,22,24,38)(15,26,19,34)(17,32,33,44)(20,28,35,43)(21,37,25,40)(23,39,36,41)(27,42,31,29)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=f,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,e^b=eh,h^b=hi,e^d=eh,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411340);