Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17405 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#537 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
659 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 659 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,19,14,18,12,17)(11,15,20,23,21,16,22,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,22)(13,20)(14,21)(16,18)(17,24)(19,23);(3,8)(5,7)(9,11)(10,16)(12,20)(13,22)(14,21)(15,18)(17,24)(19,23);(2,6)(5,7)(9,12,14,13)(10,17,18,19)(11,20,21,22)(15,23,16,24)Polycyclic presentation:
2^8|a^2=g,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139724, #139725, #139834, … |
| maximal normal subgroup of | #139724, #139725, #139834, … |
| maximal quotient of | #213512, #213513, #213514, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14745);