Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×C4×C22⋊C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3638 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)(7,15,17,11)(8,12)(9,14)(10,13)(16,27,29,22)(18,28,31,23)(19,25)(20,24)(21,26)(30,35,36,33)(32,34);(5,7,9,18,10,17,21,31)(6,11,14,23,13,15,26,28)(8,16,19,30,20,29,32,36)(12,22,25,33,24,27,34,35);(2,4);(5,8,9,19,10,20,21,32)(6,12,14,25,13,24,26,34)(7,16,18,30,17,29,31,36)(11,22,23,33,15,27,28,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,g^b=gi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72239830, #72256910, #72256921, … |
| maximal normal subgroup of | #72239830, #72256910, #72256921, … |
| maximal quotient of | #72308415, #72317350, #72317376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261999);