Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #368884 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14023 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1637 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,25)(3,14,17,9,18,32,26,31)(4,10,20,36,21,35,28,34)(5,22,23,11,24,38,27,37)(15,39,44,53,43,56,42,29)(16,30,47,54,46,55,45,40)(19,48,51,57,50,58,49,33)(41,52,60,59,61,64,62,63);(1,3,7,17,8,18,6,26)(2,9,12,32,13,31,25,14)(4,15,20,44,21,43,28,42)(5,16,23,47,24,46,27,45)(10,29,36,39,35,53,34,56)(11,30,38,54,37,55,22,40)(19,41,51,60,50,61,49,62)(33,52,48,59,57,64,58,63);(1,4)(2,10,13,35)(3,15,18,43)(5,19)(6,20)(7,28)(8,21)(9,29)(11,33,37,57)(12,34,25,36)(14,39)(16,41,46,61)(17,42,26,44)(22,48,38,58)(23,49)(24,50)(27,51)(30,52)(31,53)(32,56)(40,59)(45,60,47,62)(54,63)(55,64);(1,5)(2,11)(3,16)(4,19)(6,27)(7,23)(8,24)(9,30)(10,33)(12,38)(13,37)(14,40)(15,41)(17,47)(18,46)(20,51)(21,50)(22,25)(26,45)(28,49)(29,52)(31,55)(32,54)(34,58)(35,57)(36,48)(39,59)(42,62)(43,61)(44,60)(53,64)(56,63)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276082);