Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | Aut(G) = #72316158 (order 4096) · Inn(G) = #466 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #466 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| socle | C2 | C2×#466 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
321 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 321 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,15)(4,22)(5,11)(6,28)(7,13)(8,14)(9,19)(10,25)(12,16)(17,45)(18,40)(20,41)(21,42)(23,35)(24,55)(26,38)(27,37)(29,36)(30,57)(31,50)(32,43)(33,51)(34,56)(39,48)(44,53)(46,61)(47,52)(49,62)(54,60)(58,63)(59,64);(1,3,16,43,7,20,29,51,8,21,27,50,5,18,6,19)(2,9,28,40,11,31,37,42,14,33,36,41,13,32,12,15)(4,17,44,57,25,48,56,64,26,49,54,63,23,46,24,47)(10,30,53,45,22,52,55,61,35,58,60,62,38,59,34,39);(1,4)(2,10)(3,17)(5,23)(6,24)(7,25)(8,26)(9,30)(11,22)(12,34)(13,38)(14,35)(15,39)(16,44)(18,46)(19,47)(20,48)(21,49)(27,54)(28,53)(29,56)(31,52)(32,59)(33,58)(36,60)(37,55)(40,45)(41,62)(42,61)(43,57)(50,63)(51,64);(1,5,8,7)(2,11,14,13)(3,18,21,20)(4,23,26,25)(6,27,29,16)(9,31,33,32)(10,22,35,38)(12,28,37,36)(15,40,42,41)(17,46,49,48)(19,50,51,43)(24,54,56,44)(30,52,58,59)(34,53,55,60)(39,45,61,62)(47,63,64,57)Polycyclic presentation:
2^8|b^2=eg,d^2=h,e^2=gh,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #152507, #152517, #152640, … |
| maximal normal subgroup of | #152507, #152517, #152640, … |
| maximal quotient of | #490467, #490471, #490477, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26605);