Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33691 |
| automorphism | Aut(G) = #72305500 (order 2048) · Inn(G) = #3109 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #3109 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | #575 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
256 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 93 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:
(1,2,7,12)(3,8,14,30)(4,17,19,52)(5,22,25,57)(6,28,26,11)(9,33,35,20)(10,37,40,27)(13,41,43,51)(15,46,23,56)(16,49,45,32)(18,39,21,55)(24,34,53,36)(29,54,61,44)(31,59,38,48)(42,64,50,62)(47,63,58,60);(1,3,7,14)(2,8,12,30)(4,13,19,43)(5,23,25,15)(6,16,26,45)(9,29,35,61)(10,38,40,31)(11,32,28,49)(17,51,52,41)(18,50,21,42)(20,54,33,44)(22,56,57,46)(24,58,53,47)(27,59,37,48)(34,63,36,60)(39,64,55,62);(1,4)(2,9)(3,13)(5,24)(6,18)(7,19)(8,29)(10,39)(11,34)(12,35)(14,43)(15,47)(16,50)(17,22)(20,27)(21,26)(23,58)(25,53)(28,36)(30,61)(31,62)(32,63)(33,37)(38,64)(40,55)(41,46)(42,45)(44,48)(49,60)(51,56)(52,57)(54,59);(3,14)(4,18,19,21)(5,25)(8,30)(9,34,35,36)(10,40)(13,42,43,50)(16,45)(17,39,52,55)(20,24,33,53)(22,57)(27,37)(29,60,61,63)(32,49)(41,64,51,62)(44,58,54,47)Polycyclic presentation:
2^8|a^2=h,b^2=h,d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^b=dh,f^b=fh,d^c=dg,e^c=eh,f^c=fg,g^c=gh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #143171, #143172, #143174, … |
| maximal normal subgroup of | #143171, #143172, #143174, … |
| maximal quotient of | #494055, #494995, #495480, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26681);