Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #95 |
| commutator | C2×C4 | C4×C16 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C22 | #1374 |
| prime radical | C23 | C22⋊C16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
156 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 128 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14)(17,18,19,24,21,25,28,36,22,26,29,37,32,38,41,46,20,23,27,33,30,34,39,43,31,35,40,44,42,45,47,48);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)(18,23)(19,27)(25,34)(26,35)(28,39)(29,40)(38,45)(41,47)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=g,g^2=h,h^2=i,b^a=bc,c^a=ci,e^a=efi,d^b=df128 conjugacy classes — view the full character table.
| maximal subgroup of | #72210594, C2×#93999 |
| maximal normal subgroup of | #72210594, C2×#93999 |
| maximal quotient of | C2×#93999 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1195);