Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | #33604 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M16 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
652 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)(5,6,11,17,12,19,35,18)(7,20,25,50,26,46,57,51)(8,14,30,43,31,45,63,44)(9,15,10,34,21,16,36,48)(13,23,38,56,40,32,39,24)(22,49,54,67,27,58,60,68)(28,42,61,64,33,47,29,62)(37,55,65,59,41,52,66,53);(5,7,21,23,11,25,36,56,12,26,9,32,35,57,10,24)(6,13,34,51,18,39,15,46,19,40,48,50,17,38,16,20)(8,27,28,52,30,60,61,53,31,22,33,55,63,54,29,59)(14,41,62,67,44,65,47,49,45,37,64,68,43,66,42,58);(2,4)(5,8)(6,14)(7,22)(9,33)(10,29)(11,30)(12,31)(13,37)(15,47)(16,42)(17,43)(18,44)(19,45)(20,49)(21,28)(23,55)(24,53)(25,54)(26,27)(32,52)(34,62)(35,63)(36,61)(38,65)(39,66)(40,41)(46,58)(48,64)(50,67)(51,68)(56,59)(57,60);(5,9,11,10,12,21,35,36)(6,15,17,34,19,16,18,48)(7,23,25,56,26,32,57,24)(8,28,30,61,31,33,63,29)(13,20,38,50,40,46,39,51)(14,42,43,64,45,47,44,62)(22,52,54,53,27,55,60,59)(37,58,65,68,41,49,66,67)Polycyclic presentation:
2^9|a^2=g,b^2=eh,d^2=g,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,c^b=ci,d^b=di,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403082);