Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137659 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| 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 | |
692 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)(5,6,9,16,10,17,8,14,11,15,24,27,23,33,25,34)(7,12,19,13,20,30,22,32,21,29,35,31,26,36,18,28);(1,2,3,4)(5,7)(6,12)(8,22)(9,19)(10,20)(11,21)(13,16)(14,32)(15,29)(17,30)(18,25)(23,26)(24,35)(27,31)(28,34)(33,36);(5,7)(6,13)(8,19)(9,22)(10,26)(11,21)(12,27)(14,30)(15,31)(16,29)(17,28)(18,24)(20,23)(25,35)(32,33)(34,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,e^b=ei,e^c=eh,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72240710, #72240725, #72240727, … |
| maximal normal subgroup of | #72240710, #72240725, #72240727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55386);