Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #397 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | C4×#397 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
945 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,27)(3,14,17,9,18,32,28,31)(4,19,23,47,22,10,21,34)(5,11,25,35,26,36,24,52)(15,39,41,45,42,37,51,29)(16,38,43,30,44,56,53,55)(20,46,50,63,49,33,48,57)(40,59,60,62,61,58,64,54);(1,3)(2,9)(4,15,22,42)(5,16)(6,28)(7,17)(8,18)(10,29,19,45)(11,30)(12,32)(13,31)(14,27)(20,40,49,61)(21,51,23,41)(24,53)(25,43)(26,44)(33,54,46,62)(34,39,47,37)(35,56)(36,55)(38,52)(48,64,50,60)(57,59,63,58);(1,4,6,21,8,22,7,23)(2,10,12,34,13,19,27,47)(3,15,17,41,18,42,28,51)(5,20,24,48,26,49,25,50)(9,29,14,37,31,45,32,39)(11,33,35,57,36,46,52,63)(16,40,43,60,44,61,53,64)(30,54,38,58,55,62,56,59);(1,5,7,25,8,26,6,24)(2,11,12,35,13,36,27,52)(3,16,17,43,18,44,28,53)(4,20,23,50,22,49,21,48)(9,30,32,56,31,55,14,38)(10,33,34,57,19,46,47,63)(15,40,41,60,42,61,51,64)(29,54,39,59,45,62,37,58)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266477);