Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6,9,16,10,17,24,34,11,15,25,32,8,14,22,31)(7,12,19,28,18,26,35,33,20,27,23,36,21,13,29,30);(1,2,4,3)(5,7)(6,12)(8,21)(9,19)(10,18)(11,20)(13,14)(15,27)(16,28)(17,26)(22,29)(23,25)(24,35)(30,31)(32,36)(33,34);(1,3,4,2)(5,7)(6,13)(8,21)(9,23)(10,18)(11,20)(12,17)(14,27)(15,26)(16,33)(19,25)(22,35)(24,29)(28,31)(30,32)(34,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,e^b=ei,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239416, #72239420, #72239428, … |
| maximal normal subgroup of | #72239416, #72239420, #72239428, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55865);