Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1007 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 1007 conjugacy classes of subgroups — too many to draw. Only its 247 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,8,7)(9,10,13,11)(12,15,16,14);(10,11)(14,15)(17,18,20,22)(19,21,23,24);(1,2,3,4)(5,7,8,6)(9,11,13,10)(12,15,16,14)(17,18)(19,21)(20,22)(23,24);(9,12,13,16)(10,14,11,15)(17,19,20,23)(18,21,22,24)Polycyclic presentation:
2^8|a^2=h,b^2=eg,c^2=fh,d^2=g,b^a=be,c^a=cf,d^a=de,c^b=cg76 conjugacy classes — view the full character table.
| maximal subgroup of | #110572, #110574, #112819, … |
| maximal normal subgroup of | #110572, #110574, #112819, … |
| maximal quotient of | #153787, #153867, #153903, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7157);