Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18316 |
| 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 | #3459 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
517 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 517 conjugacy classes of subgroups — too many to draw. Only its 115 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:
(2,5)(3,7)(6,8)(9,10,14,18)(11,19,21,24)(12,17,13,16)(15,22,23,20);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,20)(13,22)(14,21)(16,19)(17,24)(18,23);(3,7)(5,8);(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,19,23,24)Polycyclic presentation:
2^8|a^2=gh,d^2=gh,e^2=g,b^a=be,c^a=cf,d^a=dgh,e^a=eg,e^b=eg,f^b=fh,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #142514, #142515, #142530, … |
| maximal normal subgroup of | #142514, #142515, #142530, … |
| maximal quotient of | #168548, #168554, #168606, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11331);