Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1438 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2474 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)(7,17)(8,22,26,32)(9,15)(10,14)(11,16)(12,19)(13,25,31,24)(18,30,23,27)(20,28)(21,29)(33,35,36,34);(2,4)(5,7,10,21,11,20,9,19)(6,12,15,28,16,29,14,17)(8,23,24,35,26,18,25,34)(13,30,22,36,31,27,32,33);(5,8)(6,13)(7,18)(9,24)(10,25)(11,26)(12,27)(14,22)(15,32)(16,31)(17,33)(19,34)(20,23)(21,35)(28,36)(29,30)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,h^a=hi,c^b=cf,f^b=fi,e^c=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237171, #72237173, #72237177, … |
| maximal normal subgroup of | #72237171, #72237173, #72237177, … |
| maximal quotient of | #72238995, #72239021 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30809);