Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12433 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #466 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
315 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 315 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,4,3)(5,6)(7,9)(8,11)(10,12)(14,21)(15,23)(16,22)(17,28)(18,19)(24,27)(25,26);(1,3,4,2)(13,14,22,27,18,23,28,26,20,25,17,15,19,24,16,21);(6,9)(7,12)(10,11)(13,15,16,25,19,26,17,23,20,27,28,14,18,21,22,24)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=d,d^2=gi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ci,e^c=egh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72226877, #72226892, #72226895 |
| maximal normal subgroup of | #72226877, #72226892, #72226895 |
| maximal quotient of | #72226875, #72226882, #72226926 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47429);