Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24283 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#390 |
| prime radical | C22×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,11)(8,18,21,12)(9,14)(10,13)(15,31,32,24)(16,25)(17,26)(19,35,34,27)(20,30,22,28)(23,29)(33,36);(2,4)(5,7,9,17,10,16,23,33)(6,11,14,26,13,25,29,36)(8,19,20,32,21,34,22,15)(12,27,28,31,18,35,30,24);(5,8)(6,12)(7,15)(9,22)(10,21)(11,24)(13,18)(14,30)(16,32)(17,34)(19,33)(20,23)(25,31)(26,35)(27,36)(28,29);(2,4)(7,16)(11,25)(15,32)(17,33)(19,34)(24,31)(26,36)(27,35)Polycyclic presentation:
2^9|a^2=h,b^2=g,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72226665, #72226677 |
| maximal normal subgroup of | #72226665, #72226677 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268812);