Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405389 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14912 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1205 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,19,37)(4,20,24,33)(5,11,26,36)(6,13,7,12)(9,27,31,18)(10,22,34,23)(16,41,44,29)(17,40,47,55)(21,49,51,48)(25,38,28,35)(30,46,56,45)(32,53,57,50)(39,52,58,43)(42,54,63,60)(59,61,64,62);(1,3)(2,9)(4,16,24,44)(5,17)(6,27)(7,18)(8,19)(10,29,34,41)(11,30)(12,37)(13,15)(14,31)(20,39,33,58)(21,42,51,63)(22,52,23,43)(25,46)(26,47)(28,45)(32,54,57,60)(35,55)(36,56)(38,40)(48,59,49,64)(50,62,53,61);(1,4,6,22,8,24,7,23)(2,10,12,33,14,34,13,20)(3,16,18,43,19,44,27,52)(5,21,25,50,26,51,28,53)(9,29,15,39,31,41,37,58)(11,32,35,49,36,57,38,48)(17,42,45,61,47,63,46,62)(30,54,40,59,56,60,55,64);(1,5)(2,11)(3,17)(4,21)(6,25)(7,28)(8,26)(9,30)(10,32)(12,35)(13,38)(14,36)(15,40)(16,42)(18,45)(19,47)(20,48)(22,50)(23,53)(24,51)(27,46)(29,54)(31,56)(33,49)(34,57)(37,55)(39,59)(41,60)(43,61)(44,63)(52,62)(58,64)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312586);