Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187419 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #62831 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14,10,3,17,20)(4,23,27,58,29,55,25,48)(5,30,32,52,34,59,42,61)(6,35,11,22,36,49,9,16)(7,15,38,19,39,21,33,13)(12,43,46,64,18,31,56,37)(24,45,26,47,63,50,62,57)(28,51,40,54,41,60,53,44);(1,3,9,22)(2,11,16,10)(4,18,28,57)(5,19,33,59)(6,14,8,35)(7,15,32,52)(12,41,47,29)(13,38,30,34)(17,49,36,20)(21,42,61,39)(23,37,51,26)(24,54,31,58)(25,45,53,56)(27,50,40,46)(43,48,63,44)(55,64,60,62);(1,4)(2,12)(3,18)(5,31)(6,27)(7,37)(8,40)(9,28)(10,29)(11,41)(13,48)(14,50)(15,51)(16,47)(17,53)(19,58)(20,45)(21,60)(22,57)(23,52)(24,33)(25,36)(26,32)(30,44)(34,43)(35,46)(38,63)(39,64)(42,62)(49,56)(54,59)(55,61);(1,5,11,38)(2,13,22,59)(3,19,16,30)(4,24,41,43)(6,7,17,42)(8,32,36,39)(9,33,10,34)(12,44,57,58)(14,15,49,61)(18,54,47,48)(20,21,35,52)(23,56,60,50)(25,62,40,37)(26,53,64,27)(28,31,29,63)(45,55,46,51)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=hi,e^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,f^b=fi,g^b=ghi,d^c=dh,e^c=ei,f^c=fh,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94735);