Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150785 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| prime radical | C23 | #357 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
210 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 86 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:
(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(17,18,20,27,36,46,48,24,23,29,19,30,22,28,37,42,21,25,34,41,32,45,31,40,39,44,33,26,38,43,35,47);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(18,24)(19,31)(20,35)(22,32)(23,39)(25,40)(26,28)(27,46)(29,47)(30,43)(33,48)(34,37)(36,38)(41,45)(42,44);(2,7)(6,9)(8,15)(14,16)(18,25)(24,40)(26,30)(27,41)(28,43)(29,44)(42,47)(45,46)Polycyclic presentation:
2^9|a^2=e,b^2=fi,d^2=gh,e^2=fgh,g^2=hi,h^2=i,b^a=bd,c^a=cf,d^a=df,d^b=dg,e^b=efg,g^b=gh,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58189);