Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C8×D4 |
| prime radical | C2×#575 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,4)(5,6,10,16,11,15,30,39,12,17,31,40,9,28,29,54)(7,13,20,34,21,33,48,58,22,35,49,59,19,46,47,65)(8,23,26,51,27,37,53,60,24,50,52,66,25,14,36,38)(18,41,44,62,45,56,64,67,42,61,63,68,43,32,55,57);(1,2,3,4)(5,7,10,20,11,21,30,48,12,22,31,49,9,19,29,47)(6,13,16,34,15,33,39,58,17,35,40,59,28,46,54,65)(8,18,26,44,27,45,53,64,24,42,52,63,25,43,36,55)(14,32,38,57,23,41,51,62,37,56,60,67,50,61,66,68);(5,8,9,25,12,24,11,27)(6,14,15,23,17,37,28,50)(7,18,19,43,22,42,21,45)(10,26,29,36,31,52,30,53)(13,32,33,41,35,56,46,61)(16,38,39,51,40,60,54,66)(20,44,47,55,49,63,48,64)(34,57,58,62,59,67,65,68);(8,24)(14,37)(18,42)(23,50)(25,27)(26,52)(32,56)(36,53)(38,60)(41,61)(43,45)(44,63)(51,66)(55,64)(57,67)(62,68)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=f,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,d^c=di128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402297);