Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #375190 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14152 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1476 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)(3,13,17,8)(4,9)(5,22,23,10)(6,25)(7,11)(12,26)(14,35,39,27)(15,28)(16,29,42,37)(18,30,45,55)(19,31,46,49)(20,33)(21,48)(24,34,51,50)(32,43)(36,40)(38,57)(41,58)(44,54)(47,63,61,56)(52,59)(53,60)(62,64);(1,3,6,16,7,17,26,42)(2,8,25,37,11,13,12,29)(4,18,43,14,20,45,21,39)(5,15,24,41,23,40,51,60)(9,30,32,35,33,55,48,27)(10,28,50,58,22,36,34,53)(19,44,61,38,46,62,47,59)(31,54,56,57,49,64,63,52);(1,4,7,20)(2,9,11,33)(3,14)(5,19,23,46)(6,21,26,43)(8,27)(10,31,22,49)(12,32,25,48)(13,35)(15,38)(16,18)(17,39)(24,47,51,61)(28,52)(29,30)(34,56,50,63)(36,57)(37,55)(40,59)(41,44)(42,45)(53,54)(58,64)(60,62);(1,5)(2,10)(3,15)(4,19)(6,24)(7,23)(8,28)(9,31)(11,22)(12,34)(13,36)(14,38)(16,41)(17,40)(18,44)(20,46)(21,47)(25,50)(26,51)(27,52)(29,53)(30,54)(32,56)(33,49)(35,57)(37,58)(39,59)(42,60)(43,61)(45,62)(48,63)(55,64)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=fhi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,h^b=hi,e^c=ei,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 282387);