Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188824 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | C2×C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
817 conjugacy classes of subgroups; only the 276 normal subgroups are shown.
This group has 817 conjugacy classes of subgroups — too many to draw. Only its 276 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:
(1,2,5,8,6,9,4,14)(3,10,13,15,11,16,12,7)(17,18)(19,27)(20,31,34,24)(21,35)(22,26)(23,30)(25,36)(28,43,32,41)(29,42)(33,40,44,47)(37,45,39,48)(38,46);(1,3,4,12,6,11,5,13)(2,7,8,10,9,15,14,16)(17,19,21,30,22,29,36,46)(18,23,25,27,26,38,35,42)(20,32,33,48,34,28,44,45)(24,39,40,43,31,37,47,41);(3,11)(7,15)(10,16)(12,13)(17,20,22,34)(18,24,26,31)(19,28,29,32)(21,33,36,44)(23,37,38,39)(25,40,35,47)(27,41,42,43)(30,45,46,48);(19,29)(23,38)(27,42)(28,32)(30,46)(37,39)(41,43)(45,48)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=i,e^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=di,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 96129);