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 | #3247 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1446 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)(3,11)(4,8)(5,18)(6,9)(7,13)(10,21)(12,29)(14,27)(15,32)(16,28)(17,25)(19,23)(20,26)(22,31)(24,30)(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);(1,3,6,14)(2,7,9,23)(4,15,17,31)(5,19,21,13)(8,24,25,29)(10,27,18,11)(12,20,30,16)(22,28,32,26)(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);(1,2)(3,7)(4,8)(5,10)(6,9)(11,13)(12,22)(14,23)(15,24)(16,26)(17,25)(18,21)(19,27)(20,28)(29,31)(30,32)(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);(1,4,6,17)(2,8,9,25)(3,12,14,30)(5,20,21,16)(7,22,23,32)(10,28,18,26)(11,29,27,24)(13,31,19,15);(2,9)(7,23)(8,25)(10,18)(11,27)(22,32)(24,29)(26,28)(34,41)(35,44)(40,55)(42,49)(43,58)(45,60)(46,56)(59,62)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=i,d^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, 6274836);