Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #361803 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13903 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1821 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,26)(3,14,18,32,17,9,28,33)(4,10,21,38,22,36,27,37)(5,23,24,39,25,11,29,40)(15,30,45,34,44,41,19,49)(16,31,48,56,46,42,47,57)(20,51,53,60,54,35,52,59)(43,61,50,58,62,55,63,64);(1,3,6,18,8,17,7,28)(2,9,26,33,12,14,13,32)(4,19,21,44,22,45,27,15)(5,16,24,47,25,46,29,48)(10,34,38,30,36,49,37,41)(11,31,40,57,23,42,39,56)(20,50,52,62,54,63,53,43)(35,58,60,55,51,64,59,61);(1,4)(2,10)(3,15)(5,20)(6,27)(7,21)(8,22)(9,30)(11,35)(12,36)(13,38)(14,41)(16,43)(17,44)(18,45)(19,28)(23,51)(24,53)(25,54)(26,37)(29,52)(31,55)(32,49)(33,34)(39,60)(40,59)(42,61)(46,62)(47,63)(48,50)(56,64)(57,58);(1,5,8,25)(2,11,12,23)(3,16,17,46)(4,20,22,54)(6,24,7,29)(9,31,14,42)(10,35,36,51)(13,39,26,40)(15,43,44,62)(18,47,28,48)(19,50,45,63)(21,52,27,53)(30,55,41,61)(32,56,33,57)(34,58,49,64)(37,59,38,60)Polycyclic presentation:
2^9|a^2=h,b^2=eg,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^b=hi,g^c=gi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269002);