Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #466 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
201 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 201 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(2,4)(5,6,11,18,13,19,12,37)(7,20,24,40,25,48,36,51)(8,26,29,58,31,59,30,45)(9,17,10,35,34,49,33,16)(14,38,41,22,43,23,42,32)(15,44,46,61,47,27,57,28)(21,39,54,64,55,66,62,65)(50,60,56,63,67,52,68,53);(1,2,3,4)(5,7,9,32,12,24,33,38,13,25,34,22,11,36,10,23)(6,14,35,20,18,42,49,51,19,43,16,48,37,41,17,40)(8,21,27,60,30,54,61,63,31,55,44,52,29,62,28,53)(15,39,58,50,46,65,59,68,47,66,45,67,57,64,26,56);(5,8,9,27,12,30,33,61,13,31,34,44,11,29,10,28)(6,15,35,58,18,46,49,59,19,47,16,45,37,57,17,26)(7,21,32,60,24,54,38,63,25,55,22,52,36,62,23,53)(14,39,20,50,42,65,51,68,43,66,48,67,41,64,40,56)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=eg,d^2=g,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,d^b=dh,f^b=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #152528, #152529, #152532, … |
| maximal normal subgroup of | #152528, #152529, #152532, … |
| maximal quotient of | #140728, #140773, #141207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6232);