Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493562 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1790 |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #62886 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,11,7,12,25,37)(3,13,17,40,16,39,47,55)(4,9,20,32,18,30,49,56)(5,21,23,36,22,10,24,34)(8,26,28,15,29,45,35,46)(14,41,44,62,42,38,60,61)(19,33,50,59,48,31,51,57)(27,52,54,43,53,64,58,63);(1,3,7,16)(2,8,12,29)(4,14,18,42)(5,15,22,46)(6,17,25,47)(9,27,30,53)(10,13,21,39)(11,35,37,28)(19,43,48,63)(20,44,49,60)(23,45,24,26)(31,41,33,38)(32,58,56,54)(34,55,36,40)(50,64,51,52)(57,61,59,62);(1,4)(2,9)(3,14)(5,19)(6,20)(7,18)(8,27)(10,33)(11,32)(12,30)(13,38)(15,43)(16,42)(17,44)(21,31)(22,48)(23,50)(24,51)(25,49)(26,52)(28,54)(29,53)(34,59)(35,58)(36,57)(37,56)(39,41)(40,61)(45,64)(46,63)(47,60)(55,62);(4,18)(9,30)(14,42)(19,48)(20,49)(27,53)(31,33)(32,56)(38,41)(43,63)(44,60)(50,51)(52,64)(54,58)(57,59)(61,62)Polycyclic presentation:
2^9|a^2=g,b^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,f^b=fhi,g^b=gh,d^c=di,e^c=eh,f^e=fi,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72303715, #72303744, #72303754, … |
| maximal normal subgroup of | #72303715, #72303744, #72303754, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400767);