Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
667 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 667 conjugacy classes of subgroups — too many to draw. Only its 135 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)(3,4)(5,6,7,8)(9,10)(11,19)(12,16)(13,22)(14,18)(15,20)(17,23)(21,24);(9,11,14,12)(10,15,18,24)(13,21,23,20)(16,22,19,17);(2,4)(9,12,14,11)(10,16,18,19)(13,20,23,21)(15,17,24,22)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=h,d^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=dh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94430, #133346, #133348, … |
| maximal normal subgroup of | #94430, #133346, #133348, … |
| maximal quotient of | #111242, #111245, #111247, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3966);