Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32947 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2847 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
639 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 639 conjugacy classes of subgroups — too many to draw. Only its 145 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)(3,13,18,36)(4,9,21,31)(5,10,23,33)(6,24,14,11)(8,25,28,17)(15,34,41,56)(16,35,43,57)(19,44,46,29)(20,47,37,30)(22,48,38,32)(26,49,52,40)(27,50,53,42)(39,59,61,55)(45,54,58,63)(51,60,62,64);(1,3,14,25,7,18,6,17)(2,8,24,36,12,28,11,13)(4,15,37,49,21,41,20,40)(5,16,38,50,23,43,22,42)(9,26,47,56,31,52,30,34)(10,27,48,57,33,53,32,35)(19,39,58,64,46,61,45,60)(29,51,63,59,44,62,54,55);(1,4)(2,9)(3,15)(5,19)(6,20)(7,21)(8,26)(10,29)(11,30)(12,31)(13,34)(14,37)(16,39)(17,40)(18,41)(22,45)(23,46)(24,47)(25,49)(27,51)(28,52)(32,54)(33,44)(35,55)(36,56)(38,58)(42,60)(43,61)(48,63)(50,64)(53,62)(57,59);(1,5)(2,10)(3,16)(4,19)(6,22)(7,23)(8,27)(9,29)(11,32)(12,33)(13,35)(14,38)(15,39)(17,42)(18,43)(20,45)(21,46)(24,48)(25,50)(26,51)(28,53)(30,54)(31,44)(34,55)(36,57)(37,58)(40,60)(41,61)(47,63)(49,64)(52,62)(56,59)Polycyclic presentation:
2^8|a^2=h,b^2=eh,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^c=gh,f^d=fh46 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25937);