Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1439 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2478 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)(5,6,9,15,11,14,10,16)(7,17,20,27,21,12,19,28)(8,22,24,30,23,31,25,13)(18,32,35,29,33,36,34,26);(2,4)(5,7,10,20,11,21,9,19)(6,12,16,28,14,17,15,27)(8,18,24,34,23,33,25,35)(13,29,22,32,30,26,31,36);(5,8)(6,13)(7,18)(9,25)(10,24)(11,23)(12,26)(14,30)(15,31)(16,22)(17,29)(19,35)(20,34)(21,33)(27,32)(28,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,g^b=gi,d^c=di,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237506, #72237507, #72237509, … |
| maximal normal subgroup of | #72237506, #72237507, #72237509, … |
| maximal quotient of | #72239070, #72239094, #72239096, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30752);