Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29902 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2622 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14265 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
933 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)(3,15,20,38,18,9,19,32)(4,21,25,35,26,39,24,10)(5,11,28,36,29,37,27,40)(16,33,45,30,44,56,22,41)(17,42,49,59,47,31,48,58)(23,50,53,62,54,61,52,34)(43,60,63,55,46,64,51,57);(1,3,6,19,8,18,7,20)(2,9,12,32,14,15,13,38)(4,22,24,44,26,45,25,16)(5,17,27,48,29,47,28,49)(10,33,21,30,35,56,39,41)(11,31,36,58,37,42,40,59)(23,51,52,46,54,63,53,43)(34,60,50,55,62,64,61,57);(1,4,6,24,8,26,7,25)(2,10,12,21,14,35,13,39)(3,16,19,22,18,44,20,45)(5,23,27,52,29,54,28,53)(9,30,32,56,15,41,38,33)(11,34,36,50,37,62,40,61)(17,46,48,63,47,43,49,51)(31,57,58,60,42,55,59,64);(1,5)(2,11)(3,17)(4,23)(6,27)(7,28)(8,29)(9,31)(10,34)(12,36)(13,40)(14,37)(15,42)(16,43)(18,47)(19,48)(20,49)(21,50)(22,51)(24,52)(25,53)(26,54)(30,55)(32,58)(33,60)(35,62)(38,59)(39,61)(41,57)(44,46)(45,63)(56,64)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,e^b=ei,g^b=gi,d^c=de,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, 287568);