Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | Aut(G) order 98304 |
| 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 | #2657 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
581 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 581 conjugacy classes of subgroups — too many to draw. Only its 155 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:
(2,4)(6,8)(9,10)(11,20)(12,22)(13,17)(14,19)(15,16)(18,21)(23,24);(1,2,3,4)(9,11,14,15)(10,16,19,20)(12,23,13,21)(17,24,22,18);(5,6,7,8)(9,12,14,13)(10,17,19,22)(11,21,15,23)(16,18,20,24);(9,13,14,12)(10,18,19,24)(11,21,15,23)(16,17,20,22)Polycyclic presentation:
2^8|b^2=e,c^2=f,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,g^a=gh,c^b=ch,g^d=gh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #33167⋊C3 |
| maximal subgroup of | #273180, #273182, #273195, … |
| maximal normal subgroup of | #273180, #273182, #273195, … |
| maximal quotient of | #351851, #351853, #351855, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26155);