Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141751 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1957 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #492 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
328 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 328 conjugacy classes of subgroups — too many to draw. Only its 114 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,15)(14,16)(17,18,22,27,21,25,23,38)(19,28,31,45,30,44,37,46)(20,32,35,40,34,48,36,26)(24,39,42,29,41,47,43,33);(2,6)(3,10)(8,13)(11,16)(17,19,20,33,23,31,36,39,21,30,34,29,22,37,35,47)(18,24,40,28,27,43,48,46,25,41,26,44,38,42,32,45);(1,3,5,11,4,10,12,16)(2,7,8,9,6,14,13,15)(18,25)(24,41)(26,40)(27,38)(28,44)(32,48)(42,43)(45,46)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,c^2=ei,d^2=g,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ch,d^b=di,f^b=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48969);