Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#535 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
734 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 734 conjugacy classes of subgroups — too many to draw. Only its 135 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) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,21)(20,24)(22,26)(23,25)(27,28);(1,2)(3,4)(6,9)(10,12)(13,15,17,22)(14,18,21,26)(16,23,24,28)(19,25,20,27);(2,4)(5,7,8,11)(6,10,9,12)(14,19)(16,24)(18,25)(20,21)(23,28)(26,27);(6,9)(10,12)(15,22)(18,26)(23,28)(25,27)Polycyclic presentation:
2^8|b^2=h,c^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139836, #139854, #346502, … |
| maximal normal subgroup of | #139836, #139854, #346502, … |
| maximal quotient of | #254709, #254716, #254723, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17001);