Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484212 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1139 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1139 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(5,8)(10,16)(11,21)(12,14)(13,24)(17,32)(18,25)(19,30)(20,31)(22,37)(23,28)(26,27)(29,33)(34,40)(35,38);(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,18)(13,25)(14,19)(15,20)(16,24)(21,33)(22,38)(23,35)(26,31)(27,30)(28,34)(29,36)(32,37)(39,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,17)(12,21)(13,22)(14,28)(15,23)(16,29)(18,33)(19,34)(20,35)(24,36)(25,38)(26,39)(27,37)(30,32)(31,40);(9,12,15,14)(10,18,20,19)(11,21,23,28)(13,24,27,26)(16,30,31,25)(17,33,35,34)(22,36,37,39)(29,32,40,38)Polycyclic presentation:
2^9|d^2=i,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391411);