Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator | C22 | C22×C42 |
| socle | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
575 conjugacy classes of subgroups; only the 351 normal subgroups are shown.
This group has 575 conjugacy classes of subgroups — too many to draw. Only its 351 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(5,6,8,7)(9,10)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24);(2,4)(9,11,12,18,13,17,19,23)(10,14,16,21,15,20,22,24);(5,7,8,6)(10,15)(14,20)(16,22)(21,24);(11,17)(14,20)(18,23)(21,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=ef,g^2=h,b^a=be,c^a=ch,d^b=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #101212, #101216, #101220, … |
| maximal normal subgroup of | #101212, #101216, #101220, … |
| maximal quotient of | #183321, #183322, #183323, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13417);