Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #104897 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #394 |
| commutator, Frattini, cosocle | C43 | C23 |
| LCS | G ▷ C43 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
285 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 285 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 512 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,18,21,25)(19,22,27,32)(20,23,29,24)(26,30,28,31)(33,34)(35,43)(36,39)(37,41)(38,42)(40,46)(44,47)(45,48);(1,3,6,13)(2,7,10,16)(4,12,5,11)(8,15,9,14)(17,19,21,27)(18,22,25,32)(20,28,29,26)(23,31,24,30)(34,39)(35,44)(36,37)(40,43)(41,42)(47,48);(2,8)(3,11)(4,5)(9,10)(12,13)(14,15)(18,23)(20,29)(22,30)(24,25)(26,28)(31,32)(33,35,36,45,38,44,37,46)(34,40,41,47,42,48,39,43)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=d,d^2=g,e^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,c^b=cf,d^b=dg,e^b=ei,f^b=fhi,e^c=egh,f^c=fghi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12111);