Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137661 |
| 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 | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
748 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)(5,6,9,13,10,16,8,21,11,15,23,31,24,33,25,14)(7,12,18,29,19,30,22,27,20,28,35,26,34,36,17,32);(1,2,3,4)(5,7)(6,12)(8,22)(9,18)(10,19)(11,20)(13,29)(14,32)(15,28)(16,30)(17,25)(21,27)(23,35)(24,34)(26,31)(33,36);(6,13)(8,23)(9,25)(10,24)(12,26)(14,16)(15,31)(17,18)(19,34)(21,33)(22,35)(27,30)(28,29)(32,36)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,e^a=ei,f^a=fi,c^b=cf,e^b=ei,e^c=eh,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72240725, #72240727, #72240736, … |
| maximal normal subgroup of | #72240725, #72240727, #72240736, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55389);