Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7503371 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×H(3,𝔽4) |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2684 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2223 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)(3,14)(4,20)(5,26)(6,30)(7,13)(8,12)(9,17)(10,24)(11,29)(15,32)(16,39)(18,34)(19,44)(21,35)(22,51)(23,37)(25,38)(27,33)(28,40)(31,41)(36,53)(42,46)(43,48)(45,58)(47,50)(49,59)(52,60)(54,56)(55,62)(57,63)(61,64);(1,3,6,17,8,19,7,18)(2,9,13,14,12,34,30,44)(4,21,24,15,25,46,23,47)(5,27,28,48,29,49,31,16)(10,35,20,50,37,42,38,32)(11,39,40,59,26,43,41,33)(22,45,54,52,55,64,53,57)(36,61,62,60,56,58,51,63);(1,4)(2,10)(3,15)(5,22)(6,23)(7,24)(8,25)(9,32)(11,36)(12,37)(13,38)(14,42)(16,45)(17,21)(18,46)(19,47)(20,30)(26,56)(27,57)(28,53)(29,55)(31,54)(33,58)(34,50)(35,44)(39,60)(40,51)(41,62)(43,63)(48,64)(49,52)(59,61);(1,5)(2,11)(3,16)(4,22)(6,31)(7,28)(8,29)(9,33)(10,36)(12,26)(13,41)(14,43)(15,45)(17,49)(18,27)(19,48)(20,51)(21,52)(23,54)(24,53)(25,55)(30,40)(32,58)(34,59)(35,60)(37,56)(38,62)(39,44)(42,63)(46,57)(47,64)(50,61);(2,12)(9,34)(10,37)(11,26)(13,30)(14,44)(20,38)(32,50)(33,59)(35,42)(36,56)(39,43)(40,41)(51,62)(58,61)(60,63)Polycyclic presentation:
2^9|b^2=g,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=df,g^c=gi,h^c=hi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411481);