Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9974 |
| automorphism | Aut(G) = C22×#72231349 (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 | #3006 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
371 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 371 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)(9,10,13,17)(11,18,21,23)(12,15,22,16)(14,19,24,20);(5,8)(9,11)(10,14)(12,19)(13,21)(15,18)(16,23)(17,24)(20,22);(1,3)(2,5)(4,7)(6,8)(10,15)(11,19)(12,22)(16,17)(18,23)(20,21)Polycyclic presentation:
2^8|a^2=g,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fh,c^b=cf,d^b=dg,f^b=fg,d^c=dh,e^c=eg,f^c=fg34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9975, #232436, #232441, … |
| maximal normal subgroup of | C2×#9975, #232436, #232441, … |
| maximal quotient of | #102430, #102432, #102441, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2963);