#504701

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×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

579 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)(7,12)(8,10)(9,11)(13,14,20,27)(15,23,31,43)(16,22,34,41)(17,35,32,44)(18,26,36,39)(19,25,29,40)(21,38,33,28)(24,30,42,37);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,29,37,21,32,18,16,20,31,19,30,33,17,36,34)(14,22,39,44,28,24,25,23,27,41,26,35,38,42,40,43);(2,4)(5,8,9,7)(6,10,11,12)(13,16,18,17,33,37,29,31,20,34,36,32,21,30,19,15)(14,23,25,42,38,44,39,41,27,43,40,24,28,35,26,22);(6,11)(10,12)(13,17,29,34,21,15,18,37,20,32,19,16,33,31,36,30)(14,24,39,23,28,41,25,35,27,42,26,43,38,22,40,44)

Polycyclic presentation:

2^9|a^2=i,b^2=ehi,c^2=e,d^2=fi,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

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, 411897);