Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C2×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C8×D4 |
| prime radical | #3150 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
716 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)(7,18,22,43,23,33,49,57,19,41,45,62,21,13,32,35)(8,14,25,37,26,36,52,59,27,38,53,60,24,50,51,66)(20,42,47,63,48,56,65,67,44,61,64,68,46,34,55,58);(5,7,9,21,12,19,11,23)(6,13,15,18,17,33,28,41)(8,20,24,46,27,44,26,48)(10,22,29,32,31,45,30,49)(14,34,36,42,38,56,50,61)(16,35,39,43,40,57,54,62)(25,47,51,55,53,64,52,65)(37,58,59,63,60,67,66,68);(1,2,3,4)(7,19)(13,33)(18,41)(20,44)(21,23)(22,45)(32,49)(34,56)(35,57)(42,61)(43,62)(46,48)(47,64)(55,65)(58,67)(63,68);(5,8,10,25,11,26,30,52,12,27,31,53,9,24,29,51)(6,14,16,37,15,36,39,59,17,38,40,60,28,50,54,66)(7,20,22,47,23,48,49,65,19,44,45,64,21,46,32,55)(13,34,35,58,18,42,43,63,33,56,57,67,41,61,62,68)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,d^2=g,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=ci128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403289);