Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
521 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 521 conjugacy classes of subgroups — too many to draw. Only its 159 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:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,21)(17,20)(18,22)(23,28)(24,26)(25,27);(6,9)(10,12)(13,15)(14,19)(16,24)(17,25)(18,23)(20,27)(21,26)(22,28);(2,4)(5,7,8,11)(6,10,9,12)(14,20)(15,23)(16,18)(19,26)(24,25)(27,28);(13,16,17,18)(14,21,20,22)(15,24,25,23)(19,26,27,28)Polycyclic presentation:
2^8|a^2=fh,c^2=e,d^2=h,g^2=h,b^a=be,c^a=cf,c^b=cg,g^b=gh,d^c=dh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #150211, #150215, #150584, … |
| maximal normal subgroup of | #150211, #150215, #150584, … |
| maximal quotient of | #242800, #242801, #242802, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16619);