Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#336 |
| prime radical | C24 | #97 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
399 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 399 conjugacy classes of subgroups — too many to draw. Only its 186 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)(3,9,10,6)(4,8)(5,7)(11,15,16,13)(12,14)(17,18,22,26,23,28,35,44,24,29,20,34,21,27,36,43)(19,30,31,47,32,42,48,40,33,45,38,46,37,25,39,41);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)(17,19)(18,25)(20,31)(21,37)(22,38)(23,32)(24,33)(26,40)(27,45)(28,30)(29,42)(34,41)(35,39)(36,48)(43,47)(44,46);(17,20,23,36,24,22,21,35)(18,26,28,44,29,34,27,43)(19,31,32,48,33,38,37,39)(25,40,30,46,42,41,45,47)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=h,d^2=i,e^2=h,f^2=g,h^2=i,b^a=bd,c^a=ci,d^a=dg,d^b=di,e^b=egi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34243);