Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #362481 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2293 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13916 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1219 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)(3,15,19,32)(4,10,24,37)(5,25,27,38)(6,12)(7,14)(8,13)(9,18,33,20)(11,26,39,28)(16,34)(17,44)(21,49)(22,51)(23,36,29,40)(30,48)(31,46)(35,54)(41,47)(42,61,62,55)(43,57)(45,56)(50,64,63,58)(52,59)(53,60);(1,3,6,18,7,19,8,20)(2,9,13,32,14,33,12,15)(4,21,29,16,24,48,23,43)(5,17,26,45,27,46,28,47)(10,34,36,49,37,57,40,30)(11,31,38,56,39,44,25,41)(22,50,54,42,53,63,52,62)(35,58,51,61,59,64,60,55);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,29)(9,30)(11,35)(12,36)(13,40)(14,37)(15,34)(17,42)(18,21)(19,43)(20,48)(25,51)(26,52)(27,53)(28,54)(31,55)(32,57)(33,49)(38,60)(39,59)(41,58)(44,61)(45,50)(46,62)(47,63)(56,64);(1,5,7,27)(2,11)(3,17,19,46)(4,22,24,53)(6,26,8,28)(9,31)(10,35)(12,25)(13,38)(14,39)(15,41)(16,42,43,62)(18,45,20,47)(21,50,48,63)(23,52,29,54)(30,55)(32,56)(33,44)(34,58)(36,51)(37,59)(40,60)(49,61)(57,64)Polycyclic presentation:
2^9|a^2=h,b^2=fg,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269677);