Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
403 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 403 conjugacy classes of subgroups — too many to draw. Only its 191 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,8,10)(7,11,12,9)(15,21)(16,23)(18,24)(20,26)(22,27)(25,28);(5,7,8,12)(6,9,10,11)(13,14)(15,18)(16,20)(17,19)(21,24)(22,25)(23,26)(27,28);(13,15,16,21,17,22,23,27)(14,18,20,24,19,25,26,28);(1,2,3,4)(14,19)(18,25)(20,26)(24,28)Polycyclic presentation:
2^8|a^2=e,b^2=e,c^2=f,d^2=g,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #142131, #142142, #300409, … |
| maximal normal subgroup of | #142131, #142142, #300409, … |
| maximal quotient of | #353055, #353057, #353059, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26306);