Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23998 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2404 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14128 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1161 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,26)(3,9,18,31,17,32,16,44)(4,19,23,10,22,35,27,34)(5,11,24,38,25,37,28,36)(14,39,41,29,20,48,45,55)(15,30,42,57,43,58,46,56)(21,50,53,59,51,47,52,33)(40,61,63,64,49,60,62,54);(1,3)(2,9)(4,20,22,14)(5,15)(6,16)(7,18)(8,17)(10,29,34,55)(11,30)(12,31)(13,32)(19,39,35,48)(21,49,51,40)(23,45,27,41)(24,42)(25,43)(26,44)(28,46)(33,54,59,64)(36,56)(37,58)(38,57)(47,60,50,61)(52,63,53,62);(1,4,7,23,8,22,6,27)(2,10,12,35,13,34,26,19)(3,14,18,41,17,20,16,45)(5,21,24,53,25,51,28,52)(9,29,31,48,32,55,44,39)(11,33,38,50,37,59,36,47)(15,40,42,63,43,49,46,62)(30,54,57,61,58,64,56,60);(1,5)(2,11)(3,15)(4,21)(6,28)(7,24)(8,25)(9,30)(10,33)(12,38)(13,37)(14,40)(16,46)(17,43)(18,42)(19,47)(20,49)(22,51)(23,53)(26,36)(27,52)(29,54)(31,57)(32,58)(34,59)(35,50)(39,60)(41,63)(44,56)(45,62)(48,61)(55,64)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 280957);