Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #466 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1808 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
179 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 179 conjugacy classes of subgroups — too many to draw. Only its 77 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 — 3 generators on 64 points:
(1,2)(3,14)(4,21,24,35)(5,25)(6,12)(7,29)(8,13)(9,17)(10,23,34,28)(11,15)(16,43,46,62)(18,39)(19,50)(20,40)(22,33,53,60)(26,36)(27,55)(30,57,58,64)(31,47)(32,48)(37,45,61,49)(38,41)(42,52,56,54)(44,51,63,59);(1,3,15,41,7,19,27,48,8,20,26,47,6,18,5,17)(2,9,25,39,12,31,36,40,13,32,55,50,29,38,11,14)(4,16,42,57,23,45,53,63,24,46,56,64,28,49,22,44)(10,30,54,62,35,59,60,61,34,58,52,43,21,51,33,37);(1,4)(2,10)(3,16)(5,22)(6,28)(7,23)(8,24)(9,30)(11,33)(12,35)(13,34)(14,37)(15,42)(17,44)(18,49)(19,45)(20,46)(21,29)(25,54)(26,56)(27,53)(31,59)(32,58)(36,60)(38,51)(39,62)(40,61)(41,57)(43,50)(47,64)(48,63)(52,55)Polycyclic presentation:
2^8|a^2=f,b^2=dg,d^2=gh,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,e^c=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #152173, #152176, #152177, … |
| maximal normal subgroup of | #152173, #152176, #152177, … |
| maximal quotient of | #134677, #134678, #134697, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5829);