Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9930 |
| automorphism | Aut(G) = C22×#72231762 (order 4096) · Inn(G) = #392 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C23×C4 | C23 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
375 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 375 conjugacy classes of subgroups — too many to draw. Only its 51 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)(4,6)(7,8)(9,10,15,20)(11,21,12,23)(13,19,14,18)(16,24,17,22);(5,8)(9,11)(10,16)(12,15)(13,24)(14,22)(17,20)(18,23)(19,21);(1,3)(2,5)(4,7)(6,8)(9,12,15,11)(10,17,20,16)(13,24,14,22)(18,23,19,21)Polycyclic presentation:
2^8|a^2=g,c^2=g,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fh,c^b=cf,d^b=dg,d^c=dh,e^c=eg31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9935, #234815, #234822, … |
| maximal normal subgroup of | C2×#9935, #234815, #234822, … |
| maximal quotient of | #102714, #102716, #102720, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2923);