Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139556 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #28143 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
411 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 411 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,9,6,8,4,14)(3,10,13,15,12,16,11,7)(17,18,21,26,23,27,22,37)(19,28,30,43,32,44,31,47)(20,33,34,39,36,48,35,25)(24,38,40,46,41,45,42,29);(1,3,6,12)(2,7)(4,11,5,13)(8,15)(9,10)(14,16)(17,19,20,29,21,30,34,38,23,32,36,46,22,31,35,45)(18,24,39,44,37,42,33,43,27,41,25,28,26,40,48,47);(2,8)(4,5)(7,15)(11,13)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,e^b=eh,d^c=dh,f^c=fh83 conjugacy classes — view the full character table.
| maximal subgroup of | #72265732, #72308613, #72308616 |
| maximal normal subgroup of | #72265732, #72308613, #72308616 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46862);