Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #99 |
| commutator | C2×C4 | C82 |
| Frattini, cosocle | C2×#405 | C22 |
| socle | C22 | #1282 |
| prime radical | C24 | #97 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×#405 ▷ C22×C4 ▷ C2 ▷ C1 | |
266 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 72 points:
(1,2,3,5,4,6,7,8)(9,10)(11,21,23,41)(12,26,27,17)(13,30)(14,20)(15,19)(16,22)(18,32)(24,49,47,42)(25,43,48,63)(28,37,53,51)(29,52,54,38)(31,57)(33,40)(34,59)(35,46)(36,45)(39,58)(44,56)(50,68,67,64)(55,62,70,69)(60,72)(61,66)(65,71);(9,11,13,24,14,25,31,50,15,23,32,47,33,48,58,67)(10,16,18,34,20,36,39,60,19,35,30,56,40,61,57,71)(12,22,28,44,29,45,55,65,27,46,53,59,54,66,70,72)(17,21,37,42,38,43,62,64,26,41,51,49,52,63,69,68)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=g,e^2=h,h^2=i,b^a=bc,c^a=cf,e^a=ei,f^a=fi,d^b=dfi,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 483);