Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24404 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24 |
| prime radical | #3490 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
656 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 656 conjugacy classes of subgroups — too many to draw. Only its 125 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,15)(13,16)(17,24)(18,22)(19,21)(20,23);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,19)(13,18)(15,21)(16,22)(17,20)(23,24);(9,12,13,20)(10,15,16,23)(11,17)(14,21)(18,19)(22,24);(2,6)(5,7)Polycyclic presentation:
2^8|b^2=e,c^2=fh,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #487621, #487625, #487629, … |
| maximal normal subgroup of | #487621, #487625, #487629, … |
| maximal quotient of | #290627, #290628, #290631, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18553);