Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
673 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 673 conjugacy classes of subgroups — too many to draw. Only its 191 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)(7,11)(8,10)(9,12)(13,14,16,20,17,19,23,26)(15,18,22,25,21,24,27,28);(5,7,8,12)(6,9,10,11)(15,21)(18,24)(22,27)(25,28);(2,4)(7,12)(9,11)(13,15)(14,18)(16,22)(17,21)(19,24)(20,25)(23,27)(26,28);(6,10)(9,11)(14,19)(15,21)(20,26)(22,27)Polycyclic presentation:
2^8|a^2=g,b^2=e,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=ceh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #112150, #112152, #112154, … |
| maximal normal subgroup of | #112150, #112152, #112154, … |
| maximal quotient of | #215136, #215137, #215138, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14779);