Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #361316 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13894 |
| 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 | |
1321 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,12,6,13)(3,9,19,33,17,31,18,32)(4,20,24,36,25,10,23,37)(5,11,27,39,26,38,28,40)(15,41,44,34,43,29,21,49)(16,45,46,57,47,30,48,56)(22,50,53,60,52,35,54,59)(42,55,51,58,62,61,63,64);(1,3)(2,9,12,31)(4,21)(5,16)(6,19)(7,18)(8,17)(10,34,20,49)(11,30,38,45)(13,33,14,32)(15,23)(22,51)(24,43)(25,44)(26,47)(27,48)(28,46)(29,37,41,36)(35,58,50,64)(39,57,40,56)(42,53)(52,63)(54,62)(55,60,61,59);(1,4,8,25)(2,10,12,20)(3,15,17,43)(5,22,26,52)(6,24,7,23)(9,29,31,41)(11,35,38,50)(13,37,14,36)(16,42,47,62)(18,44,19,21)(27,54,28,53)(30,55,45,61)(32,49,33,34)(39,60,40,59)(46,63,48,51)(56,64,57,58);(1,5)(2,11,12,38)(3,16)(4,22)(6,27)(7,28)(8,26)(9,30,31,45)(10,35,20,50)(13,39,14,40)(15,42)(17,47)(18,46)(19,48)(21,51)(23,53)(24,54)(25,52)(29,55,41,61)(32,56,33,57)(34,58,49,64)(36,59,37,60)(43,62)(44,63)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,d^2=f,g^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,h^b=hi,g^c=gi,h^c=hi,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268515);