Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#469 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
467 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 467 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:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(15,21)(16,23)(18,24)(20,26)(22,27)(25,28);(1,3,4,2)(5,7)(6,9)(8,12)(10,11)(13,14,17,19)(15,18,22,25)(16,20,23,26)(21,24,27,28);(7,12)(9,11)(13,15,16,21,17,22,23,27)(14,18,20,24,19,25,26,28);(6,10)(9,11)(14,19)(15,22)(20,26)(21,27)Polycyclic presentation:
2^8|a^2=g,b^2=gh,c^2=f,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #136192, #136220, #137400, … |
| maximal normal subgroup of | #136192, #136220, #137400, … |
| maximal quotient of | #218771, #218783, #218819, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15187);