Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352341 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2657 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,14)(3,16,21,48)(4,22,25,11)(5,12,29,43)(6,13,32,26)(7,33,35,60)(9,15,36,45)(10,37,42,18)(17,46,51,62)(19,47,56,59)(20,39,57,63)(23,58,24,31)(27,30,44,28)(34,41,61,64)(38,49,55,52)(40,54,50,53);(1,3)(2,10)(4,17)(5,18)(6,19)(7,34)(8,21)(9,20)(11,38)(12,39)(13,40)(14,42)(15,41)(16,33)(22,55)(23,47)(24,59)(25,51)(26,50)(27,52)(28,53)(29,37)(30,54)(31,46)(32,56)(35,61)(36,57)(43,63)(44,49)(45,64)(48,60)(58,62);(1,4,9,26)(2,11,15,32)(3,17,20,50)(5,27,7,23)(6,14,22,45)(8,25,36,13)(10,38,41,56)(12,28,33,31)(16,46,39,53)(18,52,34,47)(19,42,55,64)(21,51,57,40)(24,29,44,35)(30,60,58,43)(37,49,61,59)(48,62,63,54);(1,5,9,7)(2,12,15,33)(3,18,20,34)(4,23,26,27)(6,30,22,58)(8,29,36,35)(10,39,41,16)(11,31,32,28)(13,44,25,24)(14,43,45,60)(17,47,50,52)(19,54,55,62)(21,37,57,61)(38,46,56,53)(40,49,51,59)(42,63,64,48);(1,6,9,22)(2,13,15,25)(3,19,20,55)(4,14,26,45)(5,28,7,31)(8,32,36,11)(10,40,41,51)(12,27,33,23)(16,47,39,52)(17,42,50,64)(18,53,34,46)(21,56,57,38)(24,43,44,60)(29,30,35,58)(37,54,61,62)(48,59,63,49)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=h,e^2=h,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259612);