Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9239 |
| automorphism | Aut(G) = C22×#72231762 (order 4096) · Inn(G) = #386 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
354 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 75 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)(3,5,7,8)(4,6)(9,10,13,15,14,18,12,17)(11,19,21,16,22,23,20,24);(1,3)(2,5,6,8)(4,7)(10,15)(12,13)(16,19)(17,18)(20,21)(23,24);(9,11,14,22)(10,16,18,24)(12,21,13,20)(15,19,17,23)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=gh,d^2=gh,e^2=gh,f^2=gh,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,e^c=egh,f^c=fgh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9258, #198380, #198385, … |
| maximal normal subgroup of | C2×#9258, #198380, #198385, … |
| maximal quotient of | #99886, #99890, #99902, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2246);