#505205

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
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	C₂²×#135	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

720 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)(7,10)(8,9)(11,12)(13,14)(15,29)(16,24)(17,36)(18,39)(19,40)(20,27)(21,41)(22,32)(23,34)(25,37)(26,31)(28,30)(33,44)(35,43)(38,42);(1,2)(3,4)(6,9)(10,12)(13,15,19,16,21,34,18,32,20,33,37,35,30,17,31,38)(14,22,26,23,28,24,25,29,27,42,39,36,41,43,40,44);(2,4)(5,7,8,11)(6,10,9,12)(13,16,31,15,30,38,37,17,20,35,18,33,21,32,19,34)(14,23,40,22,41,44,39,43,27,36,25,42,28,29,26,24);(6,9)(10,12)(13,17,37,32,21,15,31,35,20,34,19,38,30,33,18,16)(14,24,39,44,28,42,40,23,27,43,26,29,41,22,25,36)

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^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, 412401);