Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22455 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D4×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13820 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1383 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)(3,13)(4,18,21,35)(5,10)(6,11)(7,12)(8,16)(9,20,31,25)(14,39,41,46)(15,32)(17,38)(19,47,50,59)(22,28)(23,33)(24,34)(26,36)(27,45,37,42)(29,44)(30,49,57,52)(40,61,48,64)(43,58)(51,55)(53,60)(54,63,56,62);(1,3)(2,8,12,13)(4,14)(5,22,24,15)(6,17)(7,16)(9,27,31,37)(10,32)(11,29,36,38)(18,46,35,39)(19,48,50,40)(20,42)(21,41)(23,51,53,43)(25,45)(26,44)(28,34)(30,56)(33,58)(47,61)(49,63,52,62)(54,57)(55,60)(59,64);(1,4,6,25,7,21,26,20)(2,9,11,35,12,31,36,18)(3,14,17,45,16,41,44,42)(5,19,23,52,24,50,53,49)(8,27,29,39,13,37,38,46)(10,30,33,59,34,57,60,47)(15,40,43,63,22,48,51,62)(28,54,55,61,32,56,58,64);(1,5,6,23,7,24,26,53)(2,10,11,33,12,34,36,60)(3,15,17,43,16,22,44,51)(4,19,25,52,21,50,20,49)(8,28,29,55,13,32,38,58)(9,30,35,59,31,57,18,47)(14,40,45,63,41,48,42,62)(27,54,39,61,37,56,46,64)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,d^2=fg,f^2=i,b^a=be,c^a=cf,h^a=hi,d^b=de,e^b=ei,f^c=fi,g^c=gi,h^d=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72319734, #72319742, #72319767, … |
| maximal normal subgroup of | #72319734, #72319742, #72319767, … |
| maximal quotient of | #72342997 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264176);