Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×C42 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C22 | C2×#346 |
| prime radical | C23 | #346 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
170 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(18,23,24,33)(19,27)(25,38,36,34)(26,35,37,44)(28,40)(29,41)(39,47,46,45)(42,48);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,18,19,25,20,26,28,39,21,23,29,34,30,35,42,45,22,24,27,36,31,37,40,46,32,33,41,38,43,44,48,47);(3,9)(6,12)(10,15)(13,16)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=f,e^2=g,g^2=h,h^2=i,b^a=bh,e^a=ei,c^b=cf,d^b=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72210594, C2×#149513, #72225407, … |
| maximal normal subgroup of | #72210594, C2×#149513, #72225407, … |
| maximal quotient of | C2×#149513, #72225405, #72286359, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56709);