Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9980 |
| automorphism | Aut(G) = C2×#72278229 (order 4096) · Inn(G) = #392 · Out(G) = C2×#120 |
| 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 | #3006 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
382 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 382 conjugacy classes of subgroups — too many to draw. Only its 49 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)(9,10,14,17)(11,18,20,23)(12,22,13,16)(15,21,24,19);(5,8)(9,11)(10,15)(12,21)(13,19)(14,20)(16,23)(17,24)(18,22);(1,3)(2,5)(4,7)(6,8)(11,19)(12,13)(15,23)(16,22)(18,24)(20,21)Polycyclic presentation:
2^8|a^2=g,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,f^b=fg,d^c=dh,f^c=fg28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9981, #232508, #232513, … |
| maximal normal subgroup of | C2×#9981, #232508, #232513, … |
| maximal quotient of | #102470, #102472, #102480, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2969);