Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143938 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #469 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #2076 |
| prime radical | C23 | #467 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
206 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 206 conjugacy classes of subgroups — too many to draw. Only its 70 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)(3,6)(4,7)(5,8)(9,16)(10,14)(11,13)(12,15)(17,18,20,24,32,39,46,48,22,25,34,40,21,26,33,41)(19,27,29,42,35,47,45,36,28,43,44,37,30,38,31,23);(1,3,4,10)(2,6,7,14)(5,9,12,11)(8,16,15,13)(23,36)(27,43)(37,42)(38,47);(2,7)(3,9)(5,12)(6,13)(10,11)(14,16)(17,19,21,35,22,28,32,30)(18,23,26,42,25,36,39,37)(20,31,33,29,34,45,46,44)(24,38,41,27,40,47,48,43)Polycyclic presentation:
2^9|a^2=eh,b^2=h,c^2=dg,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,f^b=fh,d^c=di,e^c=eg,f^c=fh,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53261);