Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519056 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #58997 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2470 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)(3,17)(4,22)(5,28)(6,32)(7,15)(8,14)(9,16)(10,20)(11,26)(12,30)(13,19)(18,37)(21,47)(23,49)(24,41)(25,59)(27,48)(29,39)(31,38)(33,44)(34,46)(35,43)(36,45)(40,51)(42,56)(50,54)(52,62)(53,55)(57,61)(58,64)(60,63);(1,3,7,20,9,21,8,29)(2,10,14,17,16,39,15,47)(4,23,26,18,27,51,35,52)(5,19,30,34,31,33,36,6)(11,40,22,37,43,49,48,62)(12,44,28,46,45,13,38,32)(24,56,57,60,58,25,55,50)(41,63,53,42,64,54,61,59);(1,4)(2,11)(3,18)(5,24)(6,25)(7,26)(8,35)(9,27)(10,37)(12,41)(13,42)(14,22)(15,48)(16,43)(17,49)(19,50)(20,51)(21,52)(23,29)(28,53)(30,57)(31,58)(32,54)(33,60)(34,56)(36,55)(38,61)(39,62)(40,47)(44,59)(45,64)(46,63);(1,5)(2,12)(3,19)(4,24)(6,29)(7,30)(8,36)(9,31)(10,13)(11,41)(14,28)(15,38)(16,45)(17,32)(18,50)(20,34)(21,33)(22,53)(23,25)(26,57)(27,58)(35,55)(37,42)(39,44)(40,63)(43,64)(46,47)(48,61)(49,54)(51,56)(52,60)(59,62);(1,6,8,33,9,34,7,19)(2,13,15,46,16,44,14,32)(3,5,29,36,21,31,20,30)(4,25,35,60,27,56,26,50)(10,38,47,45,39,28,17,12)(11,42,48,63,43,59,22,54)(18,24,23,55,52,58,51,57)(37,61,40,64,62,53,49,41)Polycyclic presentation:
2^9|b^2=f,e^2=fi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426254);