Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1439 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1634 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)(5,6,10,14,11,15,9,25)(7,16,20,27,19,31,18,12)(8,13,24,28,23,29,22,35)(17,32,34,26,21,30,33,36);(2,4)(5,7)(6,12,15,27)(8,21)(9,18)(10,20)(11,19)(13,26,29,36)(14,16,25,31)(17,23)(22,34)(24,33)(28,30,35,32);(5,8)(6,13)(7,17,19,21)(9,22)(10,24)(11,23)(12,26,27,36)(14,28)(15,29)(16,30,31,32)(18,33,20,34)(25,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237653, #72237660, #72237669, … |
| maximal normal subgroup of | #72237653, #72237660, #72237669, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31201);