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 | #62948 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,9,22)(3,14,31,51)(4,7,13,8)(5,20,23,38)(6,21,32,40)(10,33,45,17)(11,25,36,26)(12,29,28,30)(15,41,35,27)(16,46,47,64)(18,39,19,43)(24,37,53,55)(34,59,60,54)(42,50,63,49)(44,58,61,48)(52,56,62,57);(1,3)(2,10)(4,15)(5,16)(6,27)(7,19)(8,18)(9,31)(11,34)(12,39)(13,35)(14,21)(17,30)(20,44)(22,45)(23,47)(24,54)(25,50)(26,49)(28,43)(29,33)(32,41)(36,60)(37,46)(38,61)(40,51)(42,52)(48,57)(53,59)(55,64)(56,58)(62,63);(1,4,8,2)(3,15,18,10)(5,20,26,11)(6,28,30,40)(7,22,9,13)(12,29,21,32)(14,41,39,33)(16,44,49,34)(17,51,27,43)(19,45,31,35)(23,38,25,36)(24,55,57,62)(37,56,52,53)(42,59,46,58)(47,61,50,60)(48,63,54,64);(1,5)(2,11)(3,16)(4,20)(6,24)(7,25)(8,26)(9,23)(10,34)(12,37)(13,38)(14,42)(15,44)(17,48)(18,49)(19,50)(21,52)(22,36)(27,54)(28,55)(29,56)(30,57)(31,47)(32,53)(33,58)(35,61)(39,46)(40,62)(41,59)(43,64)(45,60)(51,63);(5,23)(11,36)(16,47)(20,38)(24,53)(25,26)(34,60)(37,55)(42,63)(44,61)(46,64)(48,58)(49,50)(52,62)(54,59)(56,57)Polycyclic presentation:
2^9|a^2=hi,c^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, 6260457);