#494083

order	512
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂³	C₂²×D₈
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

710 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,d^2=i,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,h^a=hi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401279);