Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33752 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C23 | C4×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
416 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 416 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) = 256 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,5,9,6,3,10,8)(4,11,12,15,13,7,14,16)(17,18,21,25)(19,27,29,46)(20,31,34,45)(22,36,33,26)(23,38,39,28)(24,41,43,35)(30,47,37,42)(32,44,48,40);(1,3)(2,6)(4,11)(5,8)(7,13)(9,10)(12,15)(14,16)(17,19,21,29)(18,23,25,39)(20,32,34,48)(22,37,33,30)(24,42,43,47)(26,44,36,40)(27,45,46,31)(28,41,38,35);(1,4,5,12,6,13,10,14)(2,7,9,16,3,11,8,15);(2,3)(7,11)(8,9)(15,16)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fi,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=dh,e^a=eg,g^a=gi,d^b=dh,e^b=eg,g^b=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391166);