#504004

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

598 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)(7,11)(8,9)(10,12)(13,14)(15,29)(16,33)(17,23)(18,36)(19,39)(20,40)(21,28)(22,31)(24,34)(25,38)(26,30)(27,37)(32,43)(35,42)(41,44);(1,2)(3,4)(6,9)(10,11)(13,15,19,16,20,17,18,31,21,32,38,34,37,35,30,44)(14,22,26,23,27,33,25,29,28,41,36,42,40,24,39,43);(2,4)(13,16,18,32,37,44,19,17,21,34,30,15,20,31,38,35)(14,23,25,41,40,43,26,33,28,42,39,22,27,29,36,24);(5,7,8,12)(6,10,9,11)(13,17,19,31,20,32,18,34,21,35,38,44,37,15,30,16)(14,24,26,43,27,22,25,23,28,33,36,29,40,41,39,42)

Polycyclic presentation:

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

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, 411200);