Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
421 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 421 conjugacy classes of subgroups — too many to draw. Only its 131 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) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,18,23)(15,24,19,20)(16,26,17,21)(22,27,28,25);(1,2,4,3)(6,9)(10,12)(13,15,18,19)(14,20,23,24)(16,27,17,25)(21,28,26,22);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,16,18,17)(14,21,23,26)(15,25,19,27)(20,22,24,28);(13,17,18,16)(14,22,23,28)(15,25,19,27)(20,21,24,26)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=e,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,g^a=gh,c^b=ch,g^d=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #273212, #273214, #273227, … |
| maximal normal subgroup of | #273212, #273214, #273227, … |
| maximal quotient of | #351103, #351105, #351107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26025);