Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #394 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#394 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2498 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)(5,6,10,16,11,15,9,14)(7,17,19,32,21,12,20,28)(8,22,25,31,26,30,24,13)(18,29,23,27,35,36,34,33);(5,7)(6,12)(8,23)(9,19)(10,20)(11,21)(13,29)(14,28)(15,17)(16,32)(18,25)(22,33)(24,35)(26,34)(27,30)(31,36);(2,4)(5,8)(6,13,15,31)(7,18)(9,24)(10,25)(11,26)(12,27,17,33)(14,30,16,22)(19,23)(20,34)(21,35)(28,36,32,29);(5,9,11,10)(6,14,15,16)(7,19,21,20)(8,24,26,25)(12,28,17,32)(13,30,31,22)(18,23,35,34)(27,36,33,29)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^b=hi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72316930, #72316933, #72316948, … |
| maximal normal subgroup of | #72316930, #72316933, #72316948, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265141);