Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93770 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1343 |
| commutator | C2×C22⋊C8 | C2×C4 |
| Frattini, cosocle | #1817 | C22 |
| prime radical | C24 | #102 |
| LCS | G ▷ C2×C22⋊C8 ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #1817 ◁ G | |
| LECS | G ▷ #1817 ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
210 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 28 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 — 2 generators on 48 points:
(2,6)(3,9,12,13)(7,10)(8,16)(14,15)(17,18,21,26,23,27,22,38)(19,28,30,42,32,45,31,48)(20,33,35,25,37,43,36,44)(24,34,39,47,41,46,40,29);(1,2,5,8)(3,10,12,15)(4,14,11,7)(6,9,16,13)(17,19,23,32)(18,24,27,41)(20,34,37,46)(21,31,22,30)(25,42,44,48)(26,39,38,40)(28,43,45,33)(29,36,47,35)Polycyclic presentation:
2^9|a^2=d,b^2=h,c^2=fh,d^2=hi,e^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fhi,c^b=cf,d^b=defh,e^b=eh,f^b=fh,g^b=gi,d^c=dgh,e^c=ei44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1031);