Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C22×C4 | C4×C8 |
| Frattini, cosocle | C22×#97 | C22 |
| socle | C23 | #324 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
406 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,3,4)(5,6)(7,9)(8,11)(10,12)(13,14,17,18,19,21,16,26)(15,22,23,20,25,24,27,28);(6,9,11,10)(13,15,18,24,17,23,21,28,19,25,26,22,16,27,14,20)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=i,e^2=h,h^2=i,b^a=bc,c^a=cf,e^a=efg,c^b=cg,g^b=gi,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#93375, #72214486, #72214501, … |
| maximal normal subgroup of | C2×#93375, #72214486, #72214501, … |
| maximal quotient of | #72214266, C2×#93375 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 571);