Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11799 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×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 | C2×#342 |
| prime radical | C24 | #95 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
547 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 547 conjugacy classes of subgroups — too many to draw. Only its 128 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:
(1,2,5,8,6,9,4,14)(3,10,13,15,11,16,12,7)(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);(1,3)(2,7,9,15)(4,12)(5,13)(6,11)(8,10,14,16)(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);(3,11)(7,15)(10,16)(12,13)(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,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, 35009);