Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1111 conjugacy classes of subgroups; only the 195 normal subgroups are shown.
This group has 1111 conjugacy classes of subgroups — too many to draw. Only its 195 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10)(11,18)(12,16)(13,21)(14,17)(15,19)(20,23)(22,24);(5,6)(7,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(2,4)(6,8);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)Polycyclic presentation:
2^8|d^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,c^b=cg,e^b=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #122540, #122554, #122620, … |
| maximal normal subgroup of | #122540, #122554, #122620, … |
| maximal quotient of | #232924, #232925, #232926, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16374);