Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
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 | #3054 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C22⋊C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2)(3,4)(5,6,11,18,12,19,35,16,13,20,34,50,10,17,9,31)(7,21,23,53,24,42,32,40,25,51,37,48,36,14,38,41)(8,15,28,45,29,46,26,49,30,47,59,66,27,44,33,43)(22,52,56,65,55,63,61,64,57,67,54,68,60,39,58,62);(5,7,9,32,10,36,34,23,13,25,35,38,12,24,11,37)(6,14,31,53,17,51,50,41,20,42,16,48,19,21,18,40)(8,22,33,61,27,60,59,56,30,57,26,58,29,55,28,54)(15,39,43,65,44,67,66,62,47,63,49,68,46,52,45,64);(2,4)(5,8)(6,15)(7,22)(9,33)(10,27)(11,28)(12,29)(13,30)(14,39)(16,49)(17,44)(18,45)(19,46)(20,47)(21,52)(23,56)(24,55)(25,57)(26,35)(31,43)(32,61)(34,59)(36,60)(37,54)(38,58)(40,64)(41,62)(42,63)(48,68)(50,66)(51,67)(53,65);(5,9,10,34,13,35,12,11)(6,16,17,18,20,31,19,50)(7,23,24,32,25,37,36,38)(8,26,27,28,30,33,29,59)(14,40,21,48,42,41,51,53)(15,43,44,66,47,49,46,45)(22,54,55,58,57,56,60,61)(39,62,52,65,63,64,67,68)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,d^b=de,e^b=ei,g^b=gi,d^c=di116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405486);