Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23892 |
| automorphism | Aut(G) = #72338915 (order 4096) · Inn(G) = #2403 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2403 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
805 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 805 conjugacy classes of subgroups — too many to draw. Only its 141 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)(3,14)(4,20)(5,26)(6,30)(7,12)(8,13)(9,17)(10,24)(11,27)(15,43)(16,32)(18,34)(19,33)(21,41)(22,52)(23,38)(25,37)(28,40)(29,39)(31,45)(35,49)(36,55)(42,46)(44,61)(47,58)(48,57)(50,53)(51,56)(54,60)(59,64)(62,63);(1,3)(2,9)(4,21)(5,16)(6,18)(7,17)(8,19)(10,35)(11,32)(12,33)(13,34)(14,30)(15,23)(20,31)(22,51)(24,45)(25,49)(26,42)(27,48)(28,46)(29,47)(36,59)(37,43)(38,41)(39,57)(40,58)(44,54)(50,56)(52,61)(53,64)(55,63)(60,62);(1,4)(2,10)(3,15)(5,22)(6,23)(7,24)(8,25)(9,31)(11,36)(12,37)(13,38)(14,41)(16,44)(17,21)(18,49)(19,45)(20,30)(26,50)(27,53)(28,54)(29,55)(32,56)(33,35)(34,43)(39,60)(40,52)(42,61)(46,63)(47,64)(48,51)(57,59)(58,62);(1,5,8,29)(2,11,13,40)(3,16,19,47)(4,22,25,55)(6,28,7,27)(9,32,34,58)(10,36,38,52)(12,39,30,26)(14,42,33,57)(15,44,45,64)(17,48,18,46)(20,50,37,60)(21,51,49,63)(23,54,24,53)(31,56,43,62)(35,59,41,61)Polycyclic presentation:
2^8|d^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,e^b=eh,g^b=gh,f^c=fh,g^c=gh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #151151, #151153, #151155, … |
| maximal normal subgroup of | #151151, #151153, #151155, … |
| maximal quotient of | #244526, #244527, #244528, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16881);