Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #103005 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #392 |
| commutator, Frattini, cosocle | C43 | C23 |
| LCS | G ▷ C43 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
391 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 391 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(22,31)(25,32)(33,34,38,42)(35,43,46,47)(36,39,37,41)(40,44,48,45);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,18)(19,25)(20,28)(21,23)(22,27)(24,30)(26,32)(29,31)(34,39)(36,37)(40,47)(41,42)(43,48)(44,45);(4,14)(6,15)(7,12)(10,13)(17,19)(18,22)(20,29)(21,26)(23,31)(24,32)(25,28)(27,30)(33,35,38,46)(34,40,42,48)(36,45,37,44)(39,47,41,43)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=i,d^2=g,e^2=i,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fgi,c^b=cf,d^b=dg,e^b=eghi,f^b=fgh,d^c=dh,e^c=ei,f^c=fgh50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 11839);