Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17367 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
745 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 745 conjugacy classes of subgroups — too many to draw. Only its 199 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,19,20,15)(13,16)(17,23,24,21)(18,22);(2,5)(6,8)(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24);(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23);(1,3,4,7)(2,6,5,8)(11,18)(14,22)(17,24)(21,23)Polycyclic presentation:
2^8|a^2=g,d^2=e,b^a=be,c^a=cf,f^a=fh,d^b=dh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #213833, #213834, #213835, … |
| maximal normal subgroup of | #213833, #213834, #213835, … |
| maximal quotient of | #190458, #190459, #190462, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13712);