Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6367534 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #62183 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1455 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,14)(3,15,19,30)(4,10,5,11)(6,13,7,12)(9,23,26,18)(16,25,21,32)(17,24,31,28)(20,29,22,27)(33,34,38,41)(35,39,44,54)(36,46,48,56)(37,49,51,42)(40,50,55,47)(43,57,58,64)(45,61,60,53)(52,62,63,59);(1,3)(2,9)(4,16)(5,21)(6,18)(7,23)(8,19)(10,24)(11,28)(12,15)(13,30)(14,26)(17,22)(20,31)(25,29)(27,32)(33,35)(34,39)(36,43)(37,45)(38,44)(40,52)(41,54)(42,53)(46,57)(47,59)(48,58)(49,61)(50,62)(51,60)(55,63)(56,64);(1,4,8,5)(2,10,14,11)(3,16,19,21)(6,22,7,20)(9,24,26,28)(12,29,13,27)(15,25,30,32)(17,23,31,18)(33,36,38,48)(34,40,41,55)(35,43,44,58)(37,50,51,47)(39,52,54,63)(42,56,49,46)(45,62,60,59)(53,64,61,57);(1,5,8,4)(2,11,14,10)(3,17,19,31)(6,22,7,20)(9,25,26,32)(12,29,13,27)(15,24,30,28)(16,23,21,18);(1,6,8,7)(2,12,14,13)(3,18,19,23)(4,20,5,22)(9,15,26,30)(10,27,11,29)(16,31,21,17)(24,32,28,25)(34,41)(35,44)(40,55)(42,49)(43,58)(45,60)(46,56)(59,62)Polycyclic presentation:
2^9|a^2=hi,c^2=hi,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,f^d=fh59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274855);