Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343678 |
| 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 | #62831 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(2,8,11,28)(4,19)(5,22)(9,26,32,52)(10,34,36,54)(12,29,13,23)(16,38)(17,45)(20,41)(21,49)(24,25)(27,35,55,37)(30,46,33,53)(31,57,58,64)(39,50)(42,44)(43,61)(47,48)(51,59,63,56)(60,62);(1,2,6,12)(3,9,18,33)(4,10,20,35)(5,23,22,28)(7,13,15,11)(8,25,29,24)(14,30,40,32)(16,31,44,59)(17,46,45,52)(19,37,41,36)(21,34,48,55)(26,50,53,39)(27,49,54,47)(38,56,42,58)(43,57,62,63)(51,61,64,60);(3,14)(4,20)(9,30)(10,35)(16,42)(17,39)(18,40)(19,41)(21,48)(26,46)(27,54)(31,58)(32,33)(34,55)(36,37)(38,44)(43,61)(45,50)(47,49)(51,63)(52,53)(56,59)(57,64)(60,62);(1,3,15,40)(2,9,11,32)(4,16,41,42)(5,17,25,50)(6,18,7,14)(8,26,28,52)(10,31,36,58)(12,33,13,30)(19,38,20,44)(21,43,49,61)(22,45,24,39)(23,46,29,53)(27,51,55,63)(34,57,54,64)(35,59,37,56)(47,60,48,62);(1,4)(2,10)(3,16)(5,21)(6,20)(7,19)(8,27)(9,31)(11,36)(12,35)(13,37)(14,38)(15,41)(17,43)(18,44)(22,48)(23,34)(24,47)(25,49)(26,51)(28,55)(29,54)(30,56)(32,58)(33,59)(39,60)(40,42)(45,62)(46,57)(50,61)(52,63)(53,64)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=hi,f^2=i,b^a=bf,e^a=ei,f^a=fh,f^b=fi,g^b=ghi,d^c=di,e^c=eh86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250883);