Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
723 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 723 conjugacy classes of subgroups — too many to draw. Only its 183 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(6,7)(9,10,15,11)(12,20,21,24)(13,18,14,17)(16,23,19,22);(5,6)(7,8)(9,11,15,10)(12,20,21,24)(13,17,14,18)(16,23,19,22);(2,4)(5,7,8,6)(9,12)(10,16)(11,19)(13,23)(14,22)(15,21)(17,24)(18,20);(12,21)(16,19)(20,24)(22,23)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gh,d^c=dh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #101264, #101265, #165887, … |
| maximal normal subgroup of | #101264, #101265, #165887, … |
| maximal quotient of | #238443, #238449, #238452, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16495);