Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #366639 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13995 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3020 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,28)(3,14,17,33,18,9,30,34)(4,19,23,10,22,37,29,36)(5,24,26,39,27,11,25,38)(15,42,45,31,20,49,44,40)(16,32,46,58,47,41,48,57)(21,50,53,55,52,59,54,35)(43,61,63,60,51,64,62,56);(1,3)(2,9)(4,20,22,15)(5,16)(6,17)(7,30)(8,18)(10,31,36,40)(11,32)(12,14)(13,33)(19,49,37,42)(21,51,52,43)(23,45,29,44)(24,41)(25,46)(26,48)(27,47)(28,34)(35,56,55,60)(38,57)(39,58)(50,64,59,61)(53,63,54,62);(1,4)(2,10)(3,15)(5,21)(6,29)(7,23)(8,22)(9,31)(11,35)(12,36)(13,37)(14,40)(16,43)(17,45)(18,20)(19,28)(24,55)(25,54)(26,53)(27,52)(30,44)(32,56)(33,42)(34,49)(38,50)(39,59)(41,60)(46,63)(47,51)(48,62)(57,64)(58,61);(1,5,8,27)(2,11,12,24)(3,16,18,47)(4,21,22,52)(6,26,7,25)(9,32,14,41)(10,35,36,55)(13,39,28,38)(15,43,20,51)(17,48,30,46)(19,50,37,59)(23,54,29,53)(31,56,40,60)(33,58,34,57)(42,61,49,64)(44,63,45,62)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,f^c=fi,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, 273838);