Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359679 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13867 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3586 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3662 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)(3,14)(4,20,24,10)(5,25)(6,28)(7,12)(8,13)(9,17)(11,26)(15,34,44,41)(16,32)(18,33)(19,40)(21,50,48,31)(22,52,55,56)(23,36,30,37)(27,39)(29,38)(35,53,49,54)(42,45)(43,62,64,57)(46,59)(47,58)(51,61,63,60);(1,3,8,19)(2,9,13,33)(4,21,24,48)(5,16,27,46)(6,18,7,17)(10,34,20,41)(11,32,38,59)(12,40,28,14)(15,30,44,23)(22,51,55,63)(25,42,39,58)(26,47,29,45)(31,36,50,37)(35,60,49,61)(43,53,64,54)(52,62,56,57);(1,4)(2,10)(3,15)(5,22)(6,23)(7,30)(8,24)(9,31)(11,35)(12,36)(13,20)(14,41)(16,43)(17,21)(18,48)(19,44)(25,56)(26,53)(27,55)(28,37)(29,54)(32,57)(33,50)(34,40)(38,49)(39,52)(42,61)(45,63)(46,64)(47,51)(58,60)(59,62);(1,5)(2,11)(3,16)(4,22)(6,29)(7,26)(8,27)(9,32)(10,35)(12,39)(13,38)(14,42)(15,43)(17,47)(18,45)(19,46)(20,49)(21,51)(23,54)(24,55)(25,28)(30,53)(31,57)(33,59)(34,60)(36,52)(37,56)(40,58)(41,61)(44,64)(48,63)(50,62)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=i,g^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,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, 266876);