Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11799 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| socle | C22 | C4×#99 |
| prime radical | C23 | C4×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
308 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 3 generators on 48 points:
(1,2,5,9,6,10,4,7)(3,8,13,15,11,16,12,14)(17,18)(19,27)(20,31)(21,26)(22,24)(23,29)(25,33)(28,42)(30,46)(32,40)(34,39)(35,38)(36,47)(37,43)(41,44)(45,48);(2,7,10,9)(3,11)(8,15,16,14)(12,13)(17,19,21,30,22,28,35,44)(18,23,26,36,24,37,38,48)(20,29,32,45,33,43,34,47)(25,27,39,41,31,42,40,46);(1,3,5,13,6,11,4,12)(2,8,9,15,10,16,7,14)(18,24)(19,28)(25,31)(26,38)(29,43)(30,44)(39,40)(45,47)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=g,e^2=gi,f^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,c^b=cg,d^b=dh,h^b=hi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36177);