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 | |
1831 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)(3,13,16,8)(4,9)(5,22,24,10)(6,25)(7,11)(12,27)(14,28,41,37)(15,29)(17,30,44,39)(18,46,45,31)(19,48,50,32)(20,36)(21,33)(23,35,53,51)(26,34)(38,42)(40,59)(43,60)(47,56)(49,57,52,58)(54,61)(55,62)(63,64);(1,3,6,17,7,16,27,44)(2,8,25,39,11,13,12,30)(4,18,20,41,21,45,26,14)(5,15,23,43,24,42,53,62)(9,31,36,28,33,46,34,37)(10,29,51,60,22,38,35,55)(19,47,49,61,50,63,52,40)(32,56,58,54,48,64,57,59);(1,4,7,21)(2,9,11,33)(3,14,16,41)(5,19,24,50)(6,26,27,20)(8,28,13,37)(10,32,22,48)(12,36,25,34)(15,40,42,61)(17,45,44,18)(23,52,53,49)(29,54,38,59)(30,46,39,31)(35,58,51,57)(43,63,62,47)(55,64,60,56);(1,5)(2,10)(3,15)(4,19)(6,23)(7,24)(8,29)(9,32)(11,22)(12,35)(13,38)(14,40)(16,42)(17,43)(18,47)(20,49)(21,50)(25,51)(26,52)(27,53)(28,54)(30,55)(31,56)(33,48)(34,57)(36,58)(37,59)(39,60)(41,61)(44,62)(45,63)(46,64)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=i,g^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, 269000);