Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360529 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13877 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2069 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,29)(3,14,18,38,19,9,17,33)(4,20,25,35,24,49,23,10)(5,26,27,36,28,11,30,37)(15,41,44,31,21,50,43,39)(16,32,45,59,46,40,47,58)(22,52,53,56,54,48,55,34)(42,62,63,57,51,61,64,60);(1,3,8,19)(2,9,12,14)(4,21,24,15)(5,16,28,46)(6,18,7,17)(10,31,35,39)(11,32,26,40)(13,38,29,33)(20,41,49,50)(22,51,54,42)(23,43,25,44)(27,47,30,45)(34,57,56,60)(36,59,37,58)(48,61,52,62)(53,63,55,64);(1,4,6,23,8,24,7,25)(2,10,13,20,12,35,29,49)(3,15,17,43,19,21,18,44)(5,22,27,53,28,54,30,55)(9,31,33,50,14,39,38,41)(11,34,36,48,26,56,37,52)(16,42,45,63,46,51,47,64)(32,57,58,62,40,60,59,61);(1,5)(2,11)(3,16)(4,22)(6,27)(7,30)(8,28)(9,32)(10,34)(12,26)(13,36)(14,40)(15,42)(17,45)(18,47)(19,46)(20,48)(21,51)(23,53)(24,54)(25,55)(29,37)(31,57)(33,58)(35,56)(38,59)(39,60)(41,61)(43,63)(44,64)(49,52)(50,62)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^b=hi,g^c=gi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267728);