Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #387433 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2372 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14479 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
755 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,14,6,13)(3,15,18,9,19,34,31,35)(4,20,23,10,25,38,24,39)(5,26,28,41,30,11,29,42)(16,32,45,43,46,49,21,36)(17,40,47,60,48,57,27,33)(22,52,55,37,56,50,54,58)(44,59,51,61,53,64,63,62);(1,3)(2,9)(4,21)(5,27)(6,31)(7,18)(8,19)(10,36)(11,40)(12,34)(13,15)(14,35)(16,23)(17,28)(20,49)(22,53)(24,46)(25,45)(26,57)(29,48)(30,47)(32,38)(33,41)(37,59)(39,43)(42,60)(44,56)(50,61)(51,54)(52,62)(55,63)(58,64);(1,4)(2,10,14,39)(3,16,19,46)(5,22)(6,23)(7,24)(8,25)(9,32)(11,37,26,58)(12,20,13,38)(15,43)(17,44,48,53)(18,21,31,45)(27,51,47,63)(28,54)(29,55)(30,56)(33,59)(34,36)(35,49)(40,62)(41,50,42,52)(57,61)(60,64);(1,5,6,28,8,30,7,29)(2,11,13,42,14,26,12,41)(3,17,18,27,19,48,31,47)(4,22,23,54,25,56,24,55)(9,33,34,57,35,60,15,40)(10,37,38,52,39,58,20,50)(16,44,21,51,46,53,45,63)(32,59,36,61,49,64,43,62)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=g,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^c=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294752);