Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #388 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1440 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2510 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)(5,6,10,16,11,14,9,15)(7,12,19,27,18,28,25,31)(8,20,24,13,22,30,23,29)(17,32,33,36,21,35,34,26);(2,4)(5,7)(6,12,14,28)(8,21,22,17)(9,19)(10,25)(11,18)(13,26)(15,27,16,31)(20,32)(23,34,24,33)(29,36)(30,35);(5,8)(6,13)(7,17)(9,24)(10,23)(11,22)(12,26)(14,29)(15,30)(16,20)(18,21)(19,34)(25,33)(27,35)(28,36)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,h^a=hi,c^b=cf,d^b=di,f^b=fi,g^b=gi,d^c=di,e^c=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237179, #72237182, #72237194, … |
| maximal normal subgroup of | #72237179, #72237182, #72237194, … |
| maximal quotient of | #72239057, #72239088, #72239096, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30833);