#505221

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

780 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=ei,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,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, 412417);