Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6366206 |
| 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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1479 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)(3,15,4,10)(5,11,22,28)(6,13,7,12)(9,19,25,17)(16,30,18,26)(20,32,31,24)(21,29,23,27)(33,34,38,41)(35,39,44,53)(36,46,48,56)(37,49,51,42)(40,50,55,47)(43,57,58,64)(45,61,60,54)(52,62,63,59);(1,3,8,4)(2,9,14,25)(5,20,22,31)(6,17,7,19)(10,12,15,13)(11,26,28,30)(16,23,18,21)(24,29,32,27)(33,35)(34,39)(36,43)(37,45)(38,44)(40,52)(41,53)(42,54)(46,57)(47,59)(48,58)(49,61)(50,62)(51,60)(55,63)(56,64);(1,4,8,3)(2,10,14,15)(5,18,22,16)(6,17,7,19)(9,12,25,13)(11,26,28,30)(20,21,31,23)(24,27,32,29)(33,36,38,48)(34,40,41,55)(35,43,44,58)(37,50,51,47)(39,52,53,63)(42,56,49,46)(45,62,60,59)(54,64,61,57);(1,5)(2,11)(3,16)(4,18)(6,21)(7,23)(8,22)(9,24)(10,26)(12,27)(13,29)(14,28)(15,30)(17,31)(19,20)(25,32)(34,41)(39,53)(40,55)(42,49)(46,56)(52,63)(54,61)(57,64);(1,6,8,7)(2,12,14,13)(3,17,4,19)(5,21,22,23)(9,15,25,10)(11,27,28,29)(16,31,18,20)(24,30,32,26)(35,44)(39,53)(43,58)(45,60)(52,63)(54,61)(57,64)(59,62)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=hi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,f^d=fh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6273497);