Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2069 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #396 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | C4×#396 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1033 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)(3,13)(4,18,22,9)(5,10)(6,25)(7,12)(8,16)(11,27)(14,39,19,36)(15,37)(17,38)(20,47,50,31)(21,32,26,35)(23,51)(24,34)(28,44,46,41)(29,42)(30,45)(33,53)(40,60,48,58)(43,59)(49,56,52,57)(54,62,64,61)(55,63);(1,3,6,16,7,17,27,45)(2,8,11,13,12,30,25,38)(4,19,21,44,22,14,26,41)(5,15,23,42,24,43,53,63)(9,28,32,39,18,46,35,36)(10,29,33,37,34,55,51,59)(20,48,49,62,50,40,52,61)(31,54,56,60,47,64,57,58);(1,4)(2,9)(3,14)(5,20)(6,26)(7,22)(8,28)(10,31)(11,35)(12,18)(13,36)(15,40)(16,44)(17,19)(21,27)(23,52)(24,50)(25,32)(29,54)(30,46)(33,57)(34,47)(37,58)(38,39)(41,45)(42,62)(43,48)(49,53)(51,56)(55,64)(59,60)(61,63);(1,5,6,23,7,24,27,53)(2,10,25,51,12,34,11,33)(3,15,16,42,17,43,45,63)(4,20,26,52,22,50,21,49)(8,29,38,59,30,55,13,37)(9,31,32,56,18,47,35,57)(14,40,44,62,19,48,41,61)(28,54,39,60,46,64,36,58)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=efh,e^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266583);