Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187213 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #62826 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,5,12,28,10,35,38)(3,14,18,45,17,51,26,58)(4,20,23,53,22,41,55,63)(6,9,27,34,7,13,29,44)(8,30,33,16,36,50,43,49)(11,39,42,25,37,57,61,24)(15,46,48,64,19,40,52,62)(21,31,54,59,47,32,56,60);(1,3)(2,8)(4,21)(5,26)(6,15)(7,19)(9,31)(10,36)(11,40)(12,43)(13,32)(14,41)(16,25)(17,28)(18,35)(20,51)(22,47)(23,56)(24,49)(27,52)(29,48)(30,57)(33,38)(34,60)(37,46)(39,50)(42,64)(44,59)(45,53)(54,55)(58,63)(61,62);(2,9)(4,22)(5,27)(8,31)(10,13)(11,41)(14,40)(16,49)(18,48)(20,37)(21,47)(23,57)(24,25)(26,52)(29,35)(30,56)(32,36)(39,55)(42,61)(45,58)(46,51)(50,54)(53,63)(62,64);(2,10)(3,15)(8,32)(9,13)(11,37)(12,38)(14,40)(16,47)(17,19)(18,48)(20,41)(21,49)(26,52)(30,54)(31,36)(33,60)(34,44)(42,61)(43,59)(45,62)(46,51)(50,56)(53,63)(58,64)Polycyclic presentation:
2^9|a^2=g,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94580);