Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
307 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 307 conjugacy classes of subgroups — too many to draw. Only its 119 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 — 4 generators on 24 points:
(1,2,5,7)(3,4,8,6)(9,10,13,15)(11,14,12,16)(17,18,20,21)(19,22,24,23);(1,3,5,8)(2,6,7,4)(9,11,13,12)(10,14,15,16)(17,19)(18,22)(20,24)(21,23);(9,12,13,11)(10,14,15,16)(18,21)(19,23)(22,24);(1,4,5,6)(2,3,7,8)(10,15)(14,16)Polycyclic presentation:
2^8|a^2=ef,b^2=efh,c^2=fh,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #125774, #126069, #126458, … |
| maximal normal subgroup of | #125774, #126069, #126458, … |
| maximal quotient of | #384692, #385317, #386832, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25504);