Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6367198 |
| 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 | #62218 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,13,14,24)(4,9,20,27)(5,11,6,10)(8,16,22,17)(15,29,30,25)(18,32,31,23)(19,28,21,26)(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)(2,8)(4,18)(5,17)(6,16)(7,14)(9,25)(10,13)(11,24)(12,22)(15,19)(20,31)(21,30)(23,26)(27,29)(28,32)(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);(3,14)(8,22)(13,24)(15,30)(16,17)(18,31)(23,32)(25,29)(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)(2,9)(3,15)(5,19)(6,21)(7,20)(8,23)(10,26)(11,28)(12,27)(13,29)(14,30)(16,18)(17,31)(22,32)(24,25);(1,5,7,6)(2,10,12,11)(3,16,14,17)(4,19,20,21)(8,24,22,13)(9,26,27,28)(15,18,30,31)(23,25,32,29)(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=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274495);