Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#447 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
828 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 828 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) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(9,10)(11,19)(12,22)(13,24)(14,16)(15,21)(17,23)(18,20);(5,6)(7,8)(9,11,13,21,14,12,20,23)(10,15,18,19,16,17,24,22);(1,2,3,4)(6,8)(10,16)(15,17)(18,24)(19,22);(9,12,13,23,14,11,20,21)(10,17,18,22,16,15,24,19)Polycyclic presentation:
2^8|b^2=e,c^2=fh,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg58 conjugacy classes — view the full character table.
| maximal subgroup of | #301862, #301863, #301919, … |
| maximal normal subgroup of | #301862, #301863, #301919, … |
| maximal quotient of | #298605, #298611, #298617, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18764);