Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×D8 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C2 | C22×#107 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#41 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
311 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,13,7,28)(3,14,18,39,20,40,19,48)(4,10,23,35,21,33,24,52)(5,25,26,36,27,53,15,11)(9,29,31,47,32,41,38,17)(16,37,45,60,43,58,46,63)(22,50,51,57,49,64,42,34)(30,54,56,62,55,61,59,44);(1,3,15,41,7,19,27,47,8,20,26,29,6,18,5,17)(2,9,25,39,12,31,36,40,13,32,53,48,28,38,11,14)(4,16,42,61,24,46,49,62,21,43,51,54,23,45,22,44)(10,30,50,60,35,56,57,58,33,55,64,63,52,59,34,37);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,21)(9,30)(11,34)(12,35)(13,33)(14,37)(15,42)(17,44)(18,45)(19,46)(20,43)(25,50)(26,51)(27,49)(28,52)(29,54)(31,56)(32,55)(36,57)(38,59)(39,60)(40,58)(41,61)(47,62)(48,63)(53,64);(4,21)(10,33)(16,43)(22,49)(23,24)(30,55)(34,57)(35,52)(37,58)(42,51)(44,62)(45,46)(50,64)(54,61)(56,59)(60,63)Polycyclic presentation:
2^8|a^2=f,b^2=eg,e^2=gh,f^2=h,g^2=h,b^a=be,e^a=eg,g^a=gh,d^c=dh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #33414⋊C3 |
| maximal subgroup of | #152375, #152376, #152425, … |
| maximal normal subgroup of | #152375, #152376, #152425, … |
| maximal quotient of | #483340, #483341, #483408, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26402);