Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24987 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2420 |
| 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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1223 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)(3,8,16,28)(4,9,20,31)(5,10,22,34)(6,11)(7,12)(13,35)(14,25)(15,41)(17,27,37,53)(18,29)(19,30)(21,32,24,36)(23,33,38,58)(26,42)(39,60,61,49)(40,50)(43,62)(44,52)(45,64,63,54)(46,56)(47,55)(48,57)(51,59);(1,3,13,37,6,16,7,17)(2,8,12,27,11,28,35,53)(4,18,24,14,20,44,21,40)(5,15,38,59,22,42,23,43)(9,29,32,50,31,52,36,25)(10,26,33,51,34,41,58,62)(19,45,48,39,46,63,47,61)(30,54,55,60,56,64,57,49);(1,4)(2,9)(3,14)(5,19)(6,20)(7,24)(8,25)(10,30)(11,31)(12,36)(13,21)(15,39)(16,40)(17,44)(18,37)(22,46)(23,48)(26,49)(27,52)(28,50)(29,53)(32,35)(33,57)(34,56)(38,47)(41,60)(42,61)(43,63)(45,59)(51,64)(54,62)(55,58);(1,5,6,22)(2,10)(3,15,16,42)(4,19,20,46)(7,23,13,38)(8,26)(9,30)(11,34)(12,33)(14,39,40,61)(17,43,37,59)(18,45,44,63)(21,47,24,48)(25,49)(27,51)(28,41)(29,54)(31,56)(32,55)(35,58)(36,57)(50,60)(52,64)(53,62)Polycyclic presentation:
2^9|a^2=h,b^2=fg,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,h^b=hi,g^c=gi,h^c=hi,e^d=ei,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269700);