Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6792410 |
| 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 | |
1800 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)(3,14)(4,9)(5,10)(6,12)(7,13)(8,15)(11,24)(16,42)(17,39)(18,27)(19,31)(20,37)(21,32)(22,33)(23,34)(25,36)(26,30)(28,45)(29,41)(35,51)(38,44)(40,54)(43,61)(46,52)(47,56)(48,58)(49,57)(50,55)(53,63)(59,62)(60,64);(3,15)(5,23)(6,24)(8,27)(10,34)(11,12)(14,30)(16,40)(18,26)(19,47)(20,21)(25,51)(28,42)(31,56)(32,37)(35,36)(38,54)(39,52)(41,50)(44,45)(48,49)(57,58)(59,61)(60,63);(1,3)(2,8)(4,16)(5,17)(6,26)(7,18)(9,28)(10,29)(11,27)(12,14)(13,30)(15,24)(19,43)(20,40)(21,45)(22,44)(23,50)(25,46)(31,53)(32,38)(33,54)(34,39)(35,52)(36,55)(37,42)(41,51)(47,63)(48,60)(49,62)(56,59)(57,64)(58,61);(1,4,7,22)(2,9,13,33)(3,16,18,44)(5,19,25,49)(6,21,24,20)(8,28,30,54)(10,31,36,57)(11,37,12,32)(14,38,27,42)(15,40,26,45)(17,43,46,62)(23,47,51,48)(29,53,55,64)(34,56,35,58)(39,59,52,61)(41,60,50,63);(1,5,24,51,7,25,6,23)(2,10,11,35,13,36,12,34)(3,17,15,41,18,46,26,50)(4,19,20,48,22,49,21,47)(8,29,27,52,30,55,14,39)(9,31,37,58,33,57,32,56)(16,43,40,60,44,62,45,63)(28,53,42,61,54,64,38,59)Polycyclic presentation:
2^9|d^2=i,e^2=fhi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6699610);