Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C2×#120 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C2×#1779 |
| prime radical | C2×#585 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
3139 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,7,14,27,41,8,11,28,36,5,12,25,39)(3,15,19,43,17,31,49,40,18,37,26,54,23,9,29,32)(4,10,21,35,20,34,50,60,22,33,52,59,24,38,53,62)(16,44,47,64,48,30,56,58,46,42,51,63,45,57,55,61);(1,3)(2,9)(4,16)(5,23)(6,26)(7,17)(8,18)(10,30)(11,31)(12,37)(13,40)(14,15)(19,28)(20,48)(21,51)(22,46)(24,45)(25,49)(27,29)(32,36)(33,57)(34,42)(35,61)(38,44)(39,43)(41,54)(47,52)(50,55)(53,56)(58,59)(60,64)(62,63);(1,4)(2,10)(3,16)(5,24)(6,21)(7,20)(8,22)(9,30)(11,33)(12,38)(13,35)(14,34)(15,42)(17,48)(18,46)(19,47)(23,45)(25,53)(26,51)(27,50)(28,52)(29,55)(31,57)(32,58)(36,59)(37,44)(39,62)(40,61)(41,60)(43,63)(49,56)(54,64);(2,11)(9,31)(10,33)(12,14)(13,36)(15,37)(30,57)(32,40)(34,38)(35,59)(39,41)(42,44)(43,54)(58,61)(60,62)(63,64)Polycyclic presentation:
2^9|a^2=g,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72227315, #72280969, #72280993, … |
| maximal normal subgroup of | #72280969, #72280993, #72281021, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400541);