Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343554 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62827 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(2,8,11,27)(3,14)(4,19)(5,22)(9,29,32,47)(10,34,36,53)(12,28,13,23)(15,38)(17,45)(20,43)(21,50)(24,44)(25,30,52,33)(26,35,54,37)(31,58,59,51)(39,41)(46,63)(48,49)(56,60,61,57)(62,64);(1,2)(3,9)(4,10)(5,23)(6,12)(7,13)(8,24)(11,16)(14,33)(15,30)(17,31)(18,47)(19,37)(20,35)(21,34)(22,27)(25,42)(26,48)(28,44)(29,55)(32,38)(36,43)(39,60)(40,52)(41,57)(45,59)(46,61)(49,54)(50,53)(51,63)(56,64)(58,62);(3,15)(4,20)(9,30)(10,35)(14,38)(17,45)(18,42)(19,43)(21,49)(25,47)(26,53)(29,52)(31,59)(32,33)(34,54)(36,37)(39,41)(40,55)(46,64)(48,50)(51,58)(56,61)(57,60)(62,63);(1,3,16,38)(2,9,11,32)(4,17,43,45)(5,18,44,55)(6,14,7,15)(8,25,27,52)(10,31,36,59)(12,33,13,30)(19,41,20,39)(21,46,50,64)(22,40,24,42)(23,47,28,29)(26,51,54,58)(34,61,53,56)(35,60,37,57)(48,63,49,62);(1,4,6,20)(2,10,12,35)(3,17,14,39)(5,21,22,49)(7,19,16,43)(8,26,28,53)(9,31,33,60)(11,36,13,37)(15,41,38,45)(18,46,40,62)(23,34,27,54)(24,48,44,50)(25,51,29,56)(30,57,32,59)(42,63,55,64)(47,61,52,58)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^2=h,f^2=i,b^a=bf,d^a=dh,e^a=ei,f^a=fh,f^b=fi,g^b=ghi,d^c=di,e^c=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250918);