Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29902 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2622 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14303 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
932 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,14,6,13)(3,15,20,33,18,9,19,32)(4,21,24,10,25,36,29,37)(5,11,27,38,28,39,26,40)(16,34,22,41,44,56,49,30)(17,42,48,59,46,31,47,58)(23,50,52,35,54,61,53,62)(43,60,51,57,45,64,63,55);(1,3,6,19,8,18,7,20)(2,9,12,32,14,15,13,33)(4,22,24,44,25,49,29,16)(5,17,26,47,28,46,27,48)(10,34,21,30,37,56,36,41)(11,31,38,58,39,42,40,59)(23,51,52,45,54,63,53,43)(35,60,50,55,62,64,61,57);(1,4)(2,10)(3,16)(5,23)(6,29)(7,24)(8,25)(9,30)(11,35)(12,36)(13,21)(14,37)(15,41)(17,45)(18,44)(19,49)(20,22)(26,53)(27,52)(28,54)(31,57)(32,34)(33,56)(38,61)(39,62)(40,50)(42,55)(43,46)(47,51)(48,63)(58,64)(59,60);(1,5,7,27,8,28,6,26)(2,11,12,38,14,39,13,40)(3,17,20,48,18,46,19,47)(4,23,24,52,25,54,29,53)(9,31,32,58,15,42,33,59)(10,35,36,61,37,62,21,50)(16,43,22,51,44,45,49,63)(30,55,34,60,41,57,56,64)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,e^b=ei,d^c=de,f^c=fi,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288812);