Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213788 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(13,26)(17,27)(19,34)(22,28)(24,39)(32,40)(35,37)(36,38);(9,10,14,18)(11,19,21,34)(12,16,25,33)(13,27,26,17)(15,28,30,22)(20,35,23,37)(24,40,39,32)(29,38,31,36);(2,5)(6,8)(9,11)(10,15)(12,23,25,20)(13,28)(14,21)(16,31,33,29)(17,34)(18,30)(19,27)(22,26)(24,36,39,38)(32,35,40,37);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,20)(13,24)(14,25)(15,29)(17,32)(18,33)(19,35)(21,23)(22,36)(26,39)(27,40)(28,38)(30,31)(34,37)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=f,g^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290019);