Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187756 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2138 conjugacy classes of subgroups; only the 318 normal subgroups are shown.
This group has 2138 conjugacy classes of subgroups — too many to draw. Only its 318 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 — 4 generators on 48 points:
(1,2,5,10,6,9,4,8)(3,11,12,16,13,7,14,15)(17,18)(19,27,30,39)(20,31,33,24)(21,25)(22,26)(23,29,37,34)(28,42)(32,46,48,38)(35,40)(36,43)(41,44)(45,47);(1,3,6,13)(2,7,9,11)(4,12,5,14)(8,15,10,16)(17,19,22,30)(18,23,26,37)(20,28,33,44)(21,34,35,29)(24,36,31,45)(25,39,40,27)(32,47,48,43)(38,42,46,41);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(17,20)(18,24)(19,28)(21,32)(22,33)(23,36)(25,38)(26,31)(27,41)(29,43)(30,44)(34,47)(35,48)(37,45)(39,42)(40,46);(2,9)(7,11)(8,10)(15,16)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=h,e^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,e^b=ei,g^b=ghi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95013);