Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632108 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #36702 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2162 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)(3,15)(4,10)(5,12)(6,11)(7,13)(8,14)(9,18)(16,42)(17,41)(19,32)(20,31)(21,49)(22,51)(23,34)(24,37)(25,38)(26,40)(27,39)(28,36)(29,48)(30,45)(33,44)(35,53)(43,58)(46,57)(47,56)(50,55)(52,60)(54,59)(61,62)(63,64);(1,3,8,19)(2,9,14,31)(4,21,24,48)(5,17,26,47)(6,20,7,18)(10,33,37,42)(11,15,13,32)(12,30,40,57)(16,28,44,23)(22,50,53,63)(25,46,27,45)(29,34,49,36)(35,58,51,61)(38,41,39,56)(43,54,62,52)(55,59,64,60);(1,4)(2,10)(3,16)(5,22)(6,23)(7,28)(8,24)(9,29)(11,34)(12,35)(13,36)(14,37)(15,33)(17,43)(18,21)(19,44)(20,48)(25,52)(26,53)(27,54)(30,55)(31,49)(32,42)(38,59)(39,60)(40,51)(41,58)(45,50)(46,63)(47,62)(56,61)(57,64);(2,11)(6,7)(9,15)(10,34)(12,38)(13,14)(18,20)(21,48)(23,28)(25,27)(29,33)(30,41)(31,32)(35,59)(36,37)(39,40)(42,49)(45,46)(50,63)(51,60)(52,54)(55,58)(56,57)(61,64);(1,5)(2,12)(3,17)(4,22)(6,25)(7,27)(8,26)(9,30)(10,35)(11,38)(13,39)(14,40)(15,41)(16,43)(18,45)(19,47)(20,46)(21,50)(23,52)(24,53)(28,54)(29,55)(31,57)(32,56)(33,58)(34,59)(36,60)(37,51)(42,61)(44,62)(48,63)(49,64)Polycyclic presentation:
2^9|b^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,f^a=fi,h^a=hi,c^b=ch,d^b=dfh,f^b=fi,g^b=gi,h^b=hi,h^c=hi,f^d=fi,h^d=hi,g^e=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6539306);