Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6792444 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #38904 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,15)(4,10)(5,26)(6,28)(7,13)(8,14)(9,19)(11,18)(12,30)(16,33)(17,34)(20,41)(21,36)(22,37)(23,56)(24,39)(25,40)(27,43)(29,44)(31,35)(32,46)(38,58)(42,52)(45,51)(47,50)(48,60)(49,59)(53,61)(54,63)(55,62)(57,64);(1,3)(2,9)(4,21)(5,17)(6,27)(7,19)(8,31)(10,36)(11,34)(12,43)(13,15)(14,41)(16,24)(18,51)(20,29)(22,23)(25,53)(26,45)(28,42)(30,52)(32,50)(33,39)(35,44)(37,56)(38,59)(40,61)(46,47)(48,55)(49,58)(54,57)(60,62)(63,64);(1,4)(2,10)(3,16)(5,22)(6,23)(7,25)(8,32)(9,33)(11,37)(12,38)(13,40)(14,46)(15,47)(17,48)(18,49)(19,50)(20,53)(21,31)(24,29)(26,59)(27,54)(28,56)(30,58)(34,60)(35,61)(36,41)(39,44)(42,63)(43,62)(45,64)(51,57)(52,55);(1,5,7,18)(2,11,13,26)(3,17,19,51)(4,22,25,49)(6,20,30,31)(8,27,29,52)(9,34,15,45)(10,37,40,59)(12,41,28,35)(14,42,44,43)(16,48,50,57)(21,23,53,58)(24,55,32,54)(33,60,47,64)(36,56,61,38)(39,62,46,63);(1,6,29,51,7,30,8,17)(2,12,44,34,13,28,14,45)(3,18,31,27,19,5,20,52)(4,23,24,57,25,58,32,48)(9,11,41,43,15,26,35,42)(10,38,39,60,40,56,46,64)(16,49,21,54,50,22,53,55)(33,37,36,62,47,59,61,63)Polycyclic presentation:
2^9|d^2=i,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,g^b=gi,h^b=hi,f^c=fi,h^c=hi,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6699642);