Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201978 |
| 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 | |
964 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)(3,13,15,23,17,26,16,8)(4,18,19,9,21,24,20,25)(14,22,29,27,30,32,28,31)(33,34)(35,44)(36,48,49,58)(37,52)(38,43)(39,42)(40,47)(41,51,57,50)(45,60,62,54)(46,55)(53,56)(59,61,64,63);(1,3)(2,8,12,23)(4,14)(5,16)(6,15)(7,17)(9,22,25,32)(10,26,11,13)(18,31,24,27)(19,29)(20,28)(21,30)(33,35,37,47,39,46,38,53)(34,40,43,44,42,56,52,55)(36,45,51,63,49,62,50,61)(41,54,58,59,57,60,48,64);(1,4)(2,9)(3,14)(5,20)(6,19)(7,21)(8,22)(10,18)(11,24)(12,25)(13,27)(15,29)(16,28)(17,30)(23,32)(26,31)(33,36)(34,41)(35,45)(37,51)(38,50)(39,49)(40,54)(42,57)(43,58)(44,59)(46,62)(47,63)(48,52)(53,61)(55,64)(56,60);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,13,23,26)(9,24,25,18)(14,28,30,29)(22,31,32,27)(34,42)(35,46)(36,49)(43,52)(47,53)(50,51)(54,60)(59,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=hi,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,d^c=dh,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, 109374);