Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13469 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2065 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,10,15,27,13,32,25,11,16,28,14,9,8,23,26)(7,17,20,34,21,18,35,30,22,24,36,33,19,12,29,31);(1,2,3,4)(5,7,9,19,11,22,27,21)(6,12,13,17,16,18,8,24)(10,20,23,29,28,36,32,35)(14,30,26,33,15,31,25,34);(5,8)(6,9)(7,18)(10,15)(11,13)(12,22)(14,28)(16,27)(17,21)(19,24)(20,33)(23,25)(26,32)(29,31)(30,35)(34,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=i,e^2=hi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,f^b=fi,e^c=eh,f^c=fi,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250135, #72250143, #72250188, … |
| maximal normal subgroup of | #72250135, #72250143, #72250188, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55417);