#504003

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

591 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)(13,14,21,28)(15,29,31,44)(16,24,34,41)(17,23,35,40)(18,26,32,43)(19,25,37,39)(20,38,36,27)(22,33,42,30);(1,2)(3,4)(6,9)(10,11)(13,15,19,16,20,17,32,33,21,31,37,34,36,35,18,30)(14,22,26,40,27,41,39,44,28,42,43,23,38,24,25,29);(2,4)(13,16,32,31,36,30,19,17,21,34,18,15,20,33,37,35)(14,23,39,22,38,44,26,24,28,40,25,42,27,29,43,41);(5,7,8,12)(6,10,9,11)(13,17,19,33,20,31,32,34,21,35,37,30,36,15,18,16)(14,24,26,29,27,22,39,40,28,41,43,44,38,42,25,23)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fh,d^2=eh,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, 411199);