Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405411 |
| 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 | |
1225 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)(3,15,19,40)(4,20,25,35)(5,11,28,39)(6,13,7,12)(9,29,32,18)(10,24,36,23)(16,42,44,30)(17,41,47,56)(21,33,50,59)(22,51,54,48)(26,38,27,37)(31,46,57,45)(34,52,60,53)(43,55,63,61)(49,64,62,58);(1,3)(2,9)(4,21,25,50)(5,17)(6,29)(7,18)(8,19)(10,33,36,59)(11,31)(12,40)(13,15)(14,32)(16,24,44,23)(20,30,35,42)(22,49,54,62)(26,46)(27,45)(28,47)(34,58,60,64)(37,56)(38,41)(39,57)(43,53,63,52)(48,55,51,61);(1,4,6,23,8,25,7,24)(2,10,12,20,14,36,13,35)(3,16,18,21,19,44,29,50)(5,22,26,52,28,54,27,53)(9,30,15,33,32,42,40,59)(11,34,37,48,39,60,38,51)(17,43,45,49,47,63,46,62)(31,55,41,58,57,61,56,64);(1,5)(2,11)(3,17)(4,22)(6,26)(7,27)(8,28)(9,31)(10,34)(12,37)(13,38)(14,39)(15,41)(16,43)(18,45)(19,47)(20,48)(21,49)(23,52)(24,53)(25,54)(29,46)(30,55)(32,57)(33,58)(35,51)(36,60)(40,56)(42,61)(44,63)(50,62)(59,64)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,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, 312608);