Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
597 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 597 conjugacy classes of subgroups — too many to draw. Only its 175 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:
(2,3)(7,11)(10,12)(13,14)(15,18)(16,17)(19,20)(21,22,25,24)(23,27,28,26);(1,2,4,3)(6,9)(10,12)(14,17)(18,19)(23,28)(26,27);(2,3)(5,6,8,9)(7,10,11,12)(13,14)(15,19)(16,17)(18,20)(21,23)(22,26)(24,27)(25,28);(1,3,4,2)(5,7)(6,10)(8,11)(9,12)(13,15,16,20)(14,18,17,19)(21,24,25,22)(23,27,28,26)Polycyclic presentation:
2^8|a^2=f,b^2=efgh,c^2=efg,d^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de52 conjugacy classes — view the full character table.
| maximal subgroup of | #142835, #142863, #144935, … |
| maximal normal subgroup of | #142835, #142863, #144935, … |
| maximal quotient of | #155763, #155824, #156066, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9992);