Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #394 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1442 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,15)(8,12,21,28)(9,22)(10,14)(11,17)(13,23)(16,29,19,33)(18,30)(20,35,36,27)(24,34,26,31)(25,32);(5,7)(6,11,14,25)(8,19,21,16)(9,17)(10,18)(12,26)(13,15,22,30)(20,34,36,31)(23,32)(24,28)(27,33)(29,35);(2,4)(5,8,9,20,10,21,23,36)(6,12,13,27,14,28,22,35)(7,16,17,31,18,19,32,34)(11,24,15,29,25,26,30,33)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,d^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,h^a=hi,c^b=cf,f^b=fi,e^c=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30976);