Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13374 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2023 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #11049 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
699 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,12,7,11,28,13)(3,8,17,31,16,33,47,32)(4,18,22,37,20,49,27,50)(5,23,25,10,26,39,54,38)(9,24,35,21,34,55,36,53)(14,40,42,60,19,51,48,63)(15,43,46,30,45,61,64,62)(29,44,58,41,57,56,59,52);(1,3)(2,8)(4,19,20,14)(5,15)(6,17)(7,16)(9,29,34,57)(10,30)(11,33)(12,31)(13,32)(18,40,49,51)(21,52,53,41)(22,48,27,42)(23,43)(24,56,55,44)(25,46)(26,45)(28,47)(35,58,36,59)(37,60,50,63)(38,62)(39,61)(54,64);(1,4)(2,9)(3,14)(5,24)(6,27)(7,20)(8,29)(10,37)(11,34)(12,36)(13,35)(15,44)(16,19)(17,48)(18,39)(21,54)(22,28)(23,49)(25,53)(26,55)(30,60)(31,59)(32,58)(33,57)(38,50)(40,61)(41,64)(42,47)(43,51)(45,56)(46,52)(62,63)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,f^b=fi,d^c=di,e^c=ehi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307179, #72307228, #72307252, … |
| maximal normal subgroup of | #72307179, #72307228, #72307252, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55261);