Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #108672 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1434 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
541 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 541 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9,11,6)(4,7,12,14)(5,8)(10,15)(13,16)(17,18,21,27,23,28,22,26)(19,29,31,24,33,39,32,40)(20,34,36,25,37,41,35,42)(30,38,45,43,46,48,44,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,22,32,23,33,21,31)(18,24,27,39,28,40,26,29)(20,30,36,45,37,46,35,44)(25,38,34,47,42,48,41,43);(1,4)(2,7,8,14)(3,10)(5,12)(6,13,9,15)(11,16)(17,20,21,35,23,37,22,36)(18,25,26,41,28,42,27,34)(19,30,31,44,33,46,32,45)(24,38,29,43,40,48,39,47)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,d^2=hi,e^2=hi,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,f^b=fi,e^c=ei,f^c=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 15877);