Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #368551 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14020 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1139 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)(4,18,20,48)(5,22,24,10)(6,12)(7,11)(9,21,32,40)(14,39)(15,29)(16,35,25,30)(19,47)(23,36,26,33)(27,34)(28,43)(31,50)(37,51)(38,45)(41,60,61,64)(42,55)(44,56)(46,57)(49,58)(52,53)(54,62,59,63);(1,3,6,25,7,17,27,16)(2,8,12,35,11,13,34,30)(4,14,40,43,20,42,21,51)(5,15,26,44,24,45,23,46)(9,28,48,55,32,37,18,39)(10,29,36,56,22,38,33,57)(19,41,50,63,49,61,52,62)(31,54,58,64,53,59,47,60);(1,4)(2,9,11,32)(3,14,17,42)(5,19)(6,21)(7,20)(8,28)(10,31,22,53)(12,18,34,48)(13,37)(15,41,45,61)(16,43,25,51)(23,50)(24,49)(26,52)(27,40)(29,54)(30,55)(33,58,36,47)(35,39)(38,59)(44,62,46,63)(56,60)(57,64);(1,5)(2,10)(3,15)(4,19)(6,26)(7,24)(8,29)(9,31)(11,22)(12,36)(13,38)(14,41)(16,46)(17,45)(18,47)(20,49)(21,52)(23,27)(25,44)(28,54)(30,57)(32,53)(33,34)(35,56)(37,59)(39,60)(40,50)(42,61)(43,63)(48,58)(51,62)(55,64)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=eg,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,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, 275747);