Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8746 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #381 |
| commutator, prime radical | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#381 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
859 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 859 conjugacy classes of subgroups — too many to draw. Only its 232 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) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,21,26,22,25,35,40)(19,27,29,41,30,23,34,37)(20,31,33,45,32,24,38,39)(28,36,44,47,43,42,46,48);(17,19,22,30)(18,23,25,27)(20,28,32,43)(21,34,35,29)(24,36,31,42)(26,41,40,37)(33,46,38,44)(39,48,45,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,20)(18,24,25,31)(19,28)(21,33)(22,32)(23,36,27,42)(26,39,40,45)(29,44)(30,43)(34,46)(35,38)(37,47,41,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=gh,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 97333);