Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #199552 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
952 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,11,7,12,5,10)(3,13,16,24,17,23,15,8)(4,18,20,9,21,25,19,26)(14,22,29,31,30,32,28,27)(33,34)(35,45,49,55)(36,50)(37,43)(38,42)(39,44)(40,48,56,47)(41,51)(46,60,63,54)(52,57)(53,58)(59,62,64,61);(1,3)(2,8,12,24)(4,14)(5,15)(6,16)(7,17)(9,22,26,32)(10,23,11,13)(18,27,25,31)(19,28)(20,29)(21,30)(33,35,37,47,39,49,38,48)(34,40,43,55,44,56,42,45)(36,46,51,61,52,63,53,62)(41,54,57,64,58,60,50,59);(1,4,7,21)(2,9,12,26)(3,14,17,30)(5,20,6,19)(8,22,24,32)(10,25,11,18)(13,27,23,31)(15,29,16,28)(33,36)(34,41)(35,46)(37,51)(38,53)(39,52)(40,54)(42,50)(43,57)(44,58)(45,59)(47,61)(48,62)(49,63)(55,64)(56,60);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,23,24,13)(9,18,26,25)(14,28,30,29)(22,27,32,31)(33,37,39,38)(34,42,44,43)(35,47,49,48)(36,51,52,53)(40,45,56,55)(41,50,58,57)(46,61,63,62)(54,59,60,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=h,d^2=hi,e^2=hi,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,f^c=fhi,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106822);