Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #386 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
279 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 279 conjugacy classes of subgroups — too many to draw. Only its 73 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 — 3 generators on 24 points:
(1,2,5,7,4,3,6,8)(9,10)(11,18,19,22)(12,15,20,21)(13,16)(14,17)(23,24);(1,3,5,8,4,2,6,7)(9,11,13,21,14,19,23,15)(10,12,16,22,17,20,24,18);(2,3)(7,8)(9,12)(10,15)(11,16)(13,22)(14,20)(17,21)(18,23)(19,24)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=g,f^2=h,b^a=bd,c^a=ce,e^a=eg,c^b=ch,d^b=dg,e^b=eg,f^b=fg,d^c=dg,f^c=fg52 conjugacy classes — view the full character table.
| maximal subgroup of | #94368, #94374, #94384, … |
| maximal normal subgroup of | #94368, #94374, #94384, … |
| maximal quotient of | #99541, #99563, #99566, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1949);