Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30236 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2644 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14266 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
663 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)(3,15,19,33)(4,20)(5,11)(6,12)(7,13)(8,14)(9,18,34,30)(10,24)(16,31,46,57)(17,43,49,59)(21,50,45,35)(22,38)(23,37)(25,42)(26,53)(27,39)(28,40)(29,41)(32,48,60,55)(36,51)(44,56,54,64)(47,58,63,61)(52,62);(1,3,6,18,8,19,7,30)(2,9,13,33,14,34,12,15)(4,21,24,16,25,45,23,46)(5,17,27,48,29,49,28,55)(10,35,20,31,37,50,42,57)(11,32,40,59,41,60,39,43)(22,47,51,44,52,63,26,54)(36,61,38,56,53,58,62,64);(1,4,6,24,8,25,7,23)(2,10,13,20,14,37,12,42)(3,16,18,45,19,46,30,21)(5,26,27,22,29,51,28,52)(9,31,33,50,34,57,15,35)(11,38,40,53,41,62,39,36)(17,47,48,44,49,63,55,54)(32,58,59,64,60,61,43,56);(1,5,6,27,8,29,7,28)(2,11,12,39,14,41,13,40)(3,17,18,48,19,49,30,55)(4,22,23,26,25,52,24,51)(9,32,15,43,34,60,33,59)(10,36,20,38,37,53,42,62)(16,44,21,47,46,54,45,63)(31,56,50,58,57,64,35,61)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287589);