Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #121143 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1443 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
503 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 503 conjugacy classes of subgroups — too many to draw. Only its 106 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)(4,12,14,7)(5,8)(6,11)(10,16,13,15)(17,18,22,27,23,26,21,28)(19,29,31,40,33,24,32,41)(20,34,37,43,38,44,36,25)(30,42,35,45,46,48,47,39);(1,3)(2,6,8,9)(4,13)(5,11)(7,15,12,16)(10,14)(17,19,21,31,23,33,22,32)(18,24,28,41,26,29,27,40)(20,35,36,46,38,47,37,30)(25,42,44,45,43,48,34,39);(1,4)(2,7)(3,10,11,13)(5,14)(6,15,9,16)(8,12)(17,20)(18,25,26,43)(19,30)(21,36)(22,37)(23,38)(24,39,29,45)(27,34,28,44)(31,35)(32,47)(33,46)(40,48,41,42)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=d,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=gh,f^c=fhi,g^c=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28340);