Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394341 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14716 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60398 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2542 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)(3,13)(4,9)(5,10)(6,24)(7,12)(8,16)(11,26)(14,36)(15,37)(17,38)(18,45)(19,47)(20,35)(21,32)(22,50)(23,34)(25,31)(27,43)(28,41)(29,44)(30,49)(33,52)(39,60)(40,54)(42,59)(46,58)(48,56)(51,57)(53,61)(55,63)(62,64);(1,3,6,16,7,17,26,44)(2,8,11,13,12,29,24,38)(4,18,20,43,21,14,25,40)(5,15,22,41,23,42,52,63)(9,27,31,45,32,54,35,36)(10,28,33,37,34,55,50,59)(19,46,48,62,49,39,51,61)(30,53,56,60,47,64,57,58);(1,4)(2,9)(3,14)(5,19)(6,25)(7,21)(8,27)(10,30)(11,35)(12,32)(13,36)(15,39)(16,43)(17,18)(20,26)(22,51)(23,49)(24,31)(28,53)(29,54)(33,57)(34,47)(37,58)(38,45)(40,44)(41,62)(42,46)(48,52)(50,56)(55,64)(59,60)(61,63);(1,5)(2,10)(3,15)(4,19)(6,22)(7,23)(8,28)(9,30)(11,33)(12,34)(13,37)(14,39)(16,41)(17,42)(18,46)(20,48)(21,49)(24,50)(25,51)(26,52)(27,53)(29,55)(31,56)(32,47)(35,57)(36,58)(38,59)(40,61)(43,62)(44,63)(45,60)(54,64)Polycyclic presentation:
2^9|b^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,e^c=ei,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301537);