Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6356301 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #60994 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1483 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,12)(4,9)(5,10)(6,20)(7,11)(8,16)(13,25)(14,26)(15,22)(17,23)(18,24)(19,30)(21,28)(27,31)(29,32)(33,34)(35,44)(36,42)(37,43)(38,40)(39,51)(41,46)(45,59)(47,56)(48,57)(49,54)(50,58)(52,53)(55,62)(60,63)(61,64);(1,3,5,15,7,14,6,16)(2,8,20,26,11,22,10,12)(4,13,18,27,17,29,19,28)(9,21,30,32,23,31,24,25)(34,40)(41,53)(42,54)(43,51)(44,56)(55,63)(57,58)(59,64);(1,4)(2,9)(3,13)(5,18)(6,19)(7,17)(8,21)(10,24)(11,23)(12,25)(14,29)(15,27)(16,28)(20,30)(22,31)(26,32)(33,35,38,47)(34,41,40,53)(36,45,49,61)(37,46,39,52)(42,55,54,63)(43,56,51,44)(48,60,50,62)(57,59,58,64);(3,14)(4,17)(8,22)(9,23)(12,26)(15,16)(18,19)(24,30)(33,36)(34,42)(35,45)(37,50)(38,49)(39,48)(40,54)(41,55)(43,58)(44,59)(46,62)(47,61)(51,57)(52,60)(53,63)(56,64);(1,5,7,6)(2,10,11,20)(3,15,14,16)(4,18,17,19)(8,12,22,26)(9,24,23,30)(13,27,29,28)(21,25,31,32)(33,37,38,39)(34,43,40,51)(35,46,47,52)(36,48,49,50)(41,56,53,44)(42,57,54,58)(45,60,61,62)(55,59,63,64)Polycyclic presentation:
2^9|b^2=fhi,c^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6263599);