Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6353061 |
| 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 |
| prime radical | #62958 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2647 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,9,14)(3,15,21,50)(4,22,11,43)(5,26,29,25)(6,12,31,45)(7,32,35,64)(8,13,36,46)(10,37,42,18)(16,51,38,56)(17,55,60,63)(19,49,20,39)(23,62,44,30)(24,28,34,27)(33,41,61,40)(47,54,57,53)(48,58,52,59);(1,3,8,20)(2,10,13,41)(4,16,25,54)(5,17,22,59)(6,18,7,33)(9,21,36,19)(11,38,26,53)(12,39,32,15)(14,42,46,40)(23,48,34,55)(24,63,44,52)(27,56,62,47)(28,51,30,57)(29,60,43,58)(31,37,35,61)(45,49,64,50);(1,4,9,11)(2,5,14,29)(3,16,21,38)(6,23,31,44)(7,34,35,24)(8,25,36,26)(10,17,42,60)(12,27,45,28)(13,22,46,43)(15,47,50,57)(18,48,37,52)(19,53,20,54)(30,32,62,64)(33,55,61,63)(39,56,49,51)(40,58,41,59);(1,5)(2,11)(3,17)(4,14)(6,30)(7,28)(8,22)(9,29)(10,38)(12,34)(13,26)(15,48)(16,42)(18,57)(19,58)(20,59)(21,60)(23,32)(24,45)(25,46)(27,35)(31,62)(33,51)(36,43)(37,47)(39,55)(40,54)(41,53)(44,64)(49,63)(50,52)(56,61);(1,6,8,7)(2,12,13,32)(3,18,20,33)(4,23,25,34)(5,27,22,62)(9,31,36,35)(10,39,41,15)(11,44,26,24)(14,45,46,64)(16,48,54,55)(17,56,59,47)(19,61,21,37)(28,43,30,29)(38,52,53,63)(40,50,42,49)(51,58,57,60)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=i,e^2=h,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ehi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260419);