Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18192 |
| automorphism | Aut(G) order 8192 |
| 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 | #3505 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
514 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 514 conjugacy classes of subgroups — too many to draw. Only its 121 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:
(2,6)(3,4)(7,8)(9,10,13,16)(11,14,18,21)(12,15,19,22)(17,23,24,20);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,17)(13,18)(15,20)(16,21)(19,24)(22,23);(1,3,5,4)(2,6,8,7)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,24)(21,23);(1,3,5,4)(2,7,8,6)Polycyclic presentation:
2^8|a^2=h,c^2=g,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,d^b=dg,e^b=eg,f^b=fh,e^c=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | #248469, #248476, #248481, … |
| maximal normal subgroup of | #248469, #248476, #248481, … |
| maximal quotient of | #167009, #167011, #167029, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11194);