#474226

order	512
nilpotency class	3
isoclinism	isoclinic class of #454382
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

780 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=egi,b^2=fgh,c^2=efi,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381422);