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:
(1,2,7,13,8,14,5,11)(3,15,20,44,19,43,17,42)(4,10,21,33,23,35,24,36)(6,26,25,40,27,12,28,37)(9,29,31,18,32,50,39,51)(16,45,46,62,48,41,49,61)(22,38,54,59,53,34,52,60)(30,55,57,47,58,63,56,64);(1,3,8,19)(2,9,14,32)(4,16,23,48)(5,17,7,20)(6,18,27,51)(10,30,35,58)(11,31,13,39)(12,15,26,43)(21,46,24,49)(22,47,53,64)(25,50,28,29)(33,56,36,57)(34,45,38,41)(37,42,40,44)(52,55,54,63)(59,62,60,61);(1,4)(2,10)(3,16)(5,24)(6,22)(7,21)(8,23)(9,30)(11,36)(12,38)(13,33)(14,35)(15,41)(17,49)(18,47)(19,48)(20,46)(25,54)(26,34)(27,53)(28,52)(29,55)(31,57)(32,58)(37,59)(39,56)(40,60)(42,62)(43,45)(44,61)(50,63)(51,64);(1,5,8,7)(2,11,14,13)(3,17,19,20)(4,21,23,24)(6,25,27,28)(9,31,32,39)(10,33,35,36)(12,37,26,40)(15,42,43,44)(16,46,48,49)(18,50,51,29)(22,52,53,54)(30,56,58,57)(34,59,38,60)(41,61,45,62)(47,55,64,63)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^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 | #72303722, #72303744, #72303748, … |
| maximal normal subgroup of | #72303722, #72303744, #72303748, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400769);