Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12525 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #493 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1957 |
| prime radical | C23 | #486 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
312 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 106 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:
(1,2,4,7,6,10,5,9)(3,11,13,16,12,15,14,8)(17,18)(19,27)(20,32)(21,24)(22,36)(23,30)(25,35)(26,33)(28,40)(29,42)(31,48)(34,37)(38,45)(39,44)(41,47)(43,46);(2,7,10,9)(3,12)(8,15,16,11)(13,14)(17,19,28,44,22,31,20,30,21,29,35,47,33,46,34,45)(18,23,37,48,26,39,25,27,24,38,32,43,36,41,40,42);(1,3,5,14,6,12,4,13)(2,8,7,11,10,16,9,15)(18,24)(19,29)(25,40)(26,36)(30,45)(31,46)(32,37)(44,47)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=ei,d^2=gi,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=chi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49031);