Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25798 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2433 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13902 |
| prime radical | C2×#3584 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3483 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,12,6,13)(3,15,19,39,18,9,28,32)(4,10,23,35,22,36,27,38)(5,24,26,40,25,11,29,37)(16,30,45,55,44,41,20,33)(17,31,47,57,48,42,46,56)(21,34,52,60,50,53,51,59)(43,54,49,58,62,61,63,64);(1,3,6,19,8,18,7,28)(2,9,13,32,12,15,14,39)(4,20,23,44,22,45,27,16)(5,17,26,46,25,48,29,47)(10,33,35,41,36,55,38,30)(11,31,37,56,24,42,40,57)(21,49,51,62,50,63,52,43)(34,58,59,61,53,64,60,54);(1,4)(2,10)(3,16)(5,21)(6,27)(7,23)(8,22)(9,30)(11,34)(12,36)(13,38)(14,35)(15,41)(17,43)(18,44)(19,45)(20,28)(24,53)(25,50)(26,52)(29,51)(31,54)(32,55)(33,39)(37,60)(40,59)(42,61)(46,63)(47,49)(48,62)(56,64)(57,58);(1,5)(2,11)(3,17)(4,21)(6,26)(7,29)(8,25)(9,31)(10,34)(12,24)(13,37)(14,40)(15,42)(16,43)(18,48)(19,46)(20,49)(22,50)(23,51)(27,52)(28,47)(30,54)(32,56)(33,58)(35,59)(36,53)(38,60)(39,57)(41,61)(44,62)(45,63)(55,64)Polycyclic presentation:
2^9|a^2=h,b^2=g,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei,g^c=gi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72319965, #72319992, #72320004, … |
| maximal normal subgroup of | #72319965, #72319992, #72320004, … |
| maximal quotient of | #72343124 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268849);