Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454681 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,15,16,14,20,13,19)(11,17,24,32,23,33,21,30)(12,25,27,35,29,36,28,18)(22,34,26,39,38,40,37,31);(1,3,4,7)(2,5,6,8)(10,16)(11,21)(13,15)(18,25)(19,20)(22,37)(23,24)(26,38)(27,28)(30,32)(34,40)(35,36);(3,7)(5,8)(9,11,14,23)(10,17,20,33)(12,26,29,37)(13,24,15,21)(16,30,19,32)(18,34,35,40)(22,28,38,27)(25,31,36,39);(9,12)(10,18)(11,22)(13,28)(14,29)(15,27)(16,25)(17,31)(19,36)(20,35)(21,37)(23,38)(24,26)(30,40)(32,34)(33,39)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381041);