#504276

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
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

550 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:

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

Polycyclic presentation:

2^9|a^2=efi,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^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, 411472);