Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
650 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 650 conjugacy classes of subgroups — too many to draw. Only its 137 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 — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,21,12,23)(14,20,15,19)(17,24,18,22);(2,5)(6,8)(10,16)(17,18)(19,20)(21,23);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,17,16,18)(14,24,15,22)(19,23,20,21);(9,12,13,11)(10,18,16,17)(14,24,15,22)(19,23,20,21)Polycyclic presentation:
2^8|a^2=h,c^2=e,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gh,f^c=fh,g^d=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | #248672, #248673, #248677, … |
| maximal normal subgroup of | #248672, #248673, #248677, … |
| maximal quotient of | #349911, #349912, #349914, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25988);