Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25446 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2428 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
430 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 430 conjugacy classes of subgroups — too many to draw. Only its 121 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:
(1,2,8,12,7,14,6,13)(3,15,29,34,19,33,18,9)(4,20,23,10,25,40,24,37)(5,26,30,39,28,38,27,11)(16,35,44,41,21,31,45,49)(17,46,56,58,48,32,47,42)(22,52,53,50,55,36,54,60)(43,62,63,59,51,61,64,57);(1,3,7,19)(2,9)(4,21)(5,17,28,48)(6,18,8,29)(10,35,37,31)(11,32)(12,15)(13,33)(14,34)(16,25)(20,49,40,41)(22,51)(23,45)(24,44)(26,42)(27,47,30,56)(36,59,52,57)(38,58)(39,46)(43,55)(50,62,60,61)(53,64)(54,63);(1,4)(2,10)(3,16)(5,22)(6,23)(7,25)(8,24)(9,31)(11,36)(12,20)(13,40)(14,37)(15,41)(17,43)(18,44)(19,21)(26,50)(27,53)(28,55)(29,45)(30,54)(32,57)(33,49)(34,35)(38,60)(39,52)(42,61)(46,59)(47,63)(48,51)(56,64)(58,62);(1,5,8,30,7,28,6,27)(2,11,13,38,14,39,12,26)(3,17,29,56,19,48,18,47)(4,22,24,54,25,55,23,53)(9,32,33,58,34,46,15,42)(10,36,40,60,37,52,20,50)(16,43,45,64,21,51,44,63)(31,57,49,62,35,59,41,61)Polycyclic presentation:
2^8|a^2=fh,b^2=ef,d^2=eh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #502203, #502204, #502217, … |
| maximal normal subgroup of | #502203, #502204, #502217, … |
| maximal quotient of | #288573, #288575, #288579, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18436);