#8103482

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4559 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8010678);