Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11247 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #387 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1443 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1474 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)(7,17)(8,21,24,31)(9,15)(10,14)(11,16)(12,19)(13,23,32,26)(18,27,34,36)(20,28)(22,30,33,35)(25,29);(5,7)(6,12,16,29)(8,22)(9,19)(10,25)(11,20)(13,30,32,35)(14,17,15,28)(18,23)(21,27,31,36)(24,33)(26,34);(2,4)(5,8,9,23,11,24,10,26)(6,13,14,31,16,32,15,21)(7,18,19,33,20,34,25,22)(12,27,17,30,29,36,28,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,h^a=hi,c^b=cf,f^b=fi,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31016);