Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
791 conjugacy classes of subgroups; only the 295 normal subgroups are shown.
This group has 791 conjugacy classes of subgroups — too many to draw. Only its 295 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) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,8,7)(10,13)(11,15)(17,18,20,22)(19,23,24,21);(9,10,12,13)(11,14,15,16)(17,18)(19,21)(20,22)(23,24);(1,2,3,4)(5,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24);(9,11,12,15)(10,14,13,16)(17,19,20,24)(18,21,22,23)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=fgh,d^2=e,b^a=be,c^a=cf,d^a=de,c^b=cg76 conjugacy classes — view the full character table.
| maximal subgroup of | #110233, #113111, #114850, … |
| maximal normal subgroup of | #110233, #113111, #114850, … |
| maximal quotient of | #153781, #153782, #153790, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7148);