Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
399 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 399 conjugacy classes of subgroups — too many to draw. Only its 159 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) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10)(11,16)(12,14)(13,15)(17,18,21,24,22,19,20,23);(2,4)(5,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,22,18)(20,23,21,24);(2,4)(6,8)(9,12,13,11)(10,14,15,16)(17,18)(19,22)(20,24)(21,23)Polycyclic presentation:
2^8|a^2=g,b^2=deh,c^2=de,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,c^b=cf,d^b=dh,e^b=eh,g^b=gh,d^c=dh,e^c=eh,g^c=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #94499, #97350, #97352, … |
| maximal normal subgroup of | #94499, #97350, #97352, … |
| maximal quotient of | #121622, #121633, #121637, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4360);