Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2069 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #396 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1441 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,12,18,28)(8,21)(9,16)(10,15)(11,14)(13,24)(17,33,35,30)(19,29,20,27)(22,36,34,26)(23,31)(25,32);(1,2)(3,4)(5,7,10,20,11,18,9,19)(6,12,16,29,14,28,15,27)(8,22,24,17,23,34,25,35)(13,30,21,26,32,33,31,36);(2,4)(5,8,9,24,11,23,10,25)(6,13,15,21,14,32,16,31)(7,17,19,34,18,35,20,22)(12,26,27,33,28,36,29,30)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,e^2=i,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^b=di,g^b=gi,d^c=di,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72252100, #72252106, #72252114, … |
| maximal normal subgroup of | #72252100, #72252106, #72252114, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31196);