Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201958 |
| 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 | #3192 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
971 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,10,7,11,5,21)(3,12,16,23,14,27,15,8)(4,17,19,9,20,24,18,25)(13,22,30,26,28,32,29,31)(33,34)(35,44)(36,49,52,57)(37,53)(38,42)(39,43)(40,47)(41,50,58,51)(45,60,62,54)(46,56)(48,55)(59,61,64,63);(1,3)(2,8,11,23)(4,13)(5,15)(6,16)(7,14)(9,22,25,32)(10,12,21,27)(17,31,24,26)(18,29)(19,30)(20,28)(33,35,37,47,39,48,38,46)(34,40,42,44,43,56,53,55)(36,45,50,63,52,62,51,61)(41,54,57,59,58,60,49,64);(1,4)(2,9)(3,13)(5,18)(6,19)(7,20)(8,22)(10,24)(11,25)(12,26)(14,28)(15,29)(16,30)(17,21)(23,32)(27,31)(33,36)(34,41)(35,45)(37,50)(38,51)(39,52)(40,54)(42,57)(43,58)(44,59)(46,61)(47,63)(48,62)(49,53)(55,64)(56,60);(3,14)(8,23)(12,27)(13,28)(15,16)(22,32)(26,31)(29,30)(33,37,39,38)(34,42,43,53)(35,46,48,47)(36,50,52,51)(40,55,56,44)(41,57,58,49)(45,61,62,63)(54,64,60,59)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=i,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,d^b=dh,f^b=fi,e^c=ei,f^c=fh,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109361);