Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #127 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2397 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1140 conjugacy classes of subgroups; only the 298 normal subgroups are shown.
This group has 1140 conjugacy classes of subgroups — too many to draw. Only its 298 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,15,19,28)(8,13)(9,14)(10,12)(11,18,25,22)(16,23)(17,29,32,36)(20,27)(21,26)(24,31,35,33)(30,34);(5,7,16,18,10,19,9,22)(6,11,23,15,12,25,14,28)(8,17,30,31,21,32,20,33)(13,24,34,29,26,35,27,36);(6,12)(13,26)(14,23)(27,34);(1,3,4,2)(5,8,9,20,10,21,16,30)(6,13,14,27,12,26,23,34)(7,17,22,33,19,32,18,31)(11,24,28,36,25,35,15,29)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=eh,e^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72265431, #72307328, #72307382 |
| maximal normal subgroup of | #72265431, #72307328, #72307382 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266481);