Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #488689 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #62244 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
766 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 766 conjugacy classes of subgroups — too many to draw. Only its 134 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)(3,6)(4,7)(5,8)(9,15)(10,13)(11,14)(12,16)(17,18,23,28)(19,29,33,48)(20,26,35,44)(21,25,34,45)(22,38,37,27)(24,40,43,31)(30,42,36,47)(32,41,39,46);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,19)(18,24)(20,31)(21,36)(22,39)(23,33)(25,29)(26,46)(27,47)(28,43)(30,34)(32,37)(35,40)(38,42)(41,44)(45,48);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15);(2,8)(6,13)(7,14)(12,15)(17,20,22,34,23,35,37,21)(18,25,27,44,28,45,38,26)(19,30,32,31,33,36,39,40)(24,41,42,29,43,46,47,48)Polycyclic presentation:
2^9|a^2=i,d^2=g,e^2=gi,g^2=i,b^a=be,c^a=cf,d^a=dghi,e^a=eg,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 395890);