Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22455 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
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 | D4×#95 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1847 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)(3,14,18,9)(4,20)(5,11)(6,28)(7,13)(8,12)(10,22)(15,41,44,48)(16,39,24,31)(17,32,19,40)(21,49)(23,35)(25,37)(26,53)(27,36)(29,34)(30,43,38,47)(33,52)(42,61,50,64)(45,56,46,60)(51,57)(54,58)(55,62,59,63);(1,3,7,19,8,18,6,17)(2,9,13,32,12,14,28,40)(4,15,22,43,23,44,29,47)(5,24,27,45,25,16,26,46)(10,30,35,41,34,38,20,48)(11,31,36,56,37,39,53,60)(21,50,52,63,51,42,54,62)(33,55,57,61,58,59,49,64);(1,4)(2,10)(3,15)(5,21)(6,29)(7,22)(8,23)(9,30)(11,33)(12,34)(13,35)(14,38)(16,42)(17,47)(18,44)(19,43)(20,28)(24,50)(25,51)(26,54)(27,52)(31,55)(32,41)(36,57)(37,58)(39,59)(40,48)(45,63)(46,62)(49,53)(56,61)(60,64);(1,5)(2,11)(3,16,18,24)(4,21)(6,26)(7,27)(8,25)(9,31,14,39)(10,33)(12,37)(13,36)(15,42,44,50)(17,45,19,46)(20,49)(22,52)(23,51)(28,53)(29,54)(30,55,38,59)(32,56,40,60)(34,58)(35,57)(41,61,48,64)(43,62,47,63)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,d^b=de,g^b=gi,f^c=fi,e^d=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72319726, #72319742, #72319754, … |
| maximal normal subgroup of | #72319726, #72319742, #72319754, … |
| maximal quotient of | #72343003 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263704);