Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
353 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 353 conjugacy classes of subgroups — too many to draw. Only its 165 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:
(1,2,4,7)(3,5,8,6)(9,10,12,16,13,15,19,22)(11,14,17,21,18,20,23,24);(2,5)(3,8)(6,7);(9,11,12,17,13,18,19,23)(10,14,16,21,15,20,22,24);(10,15)(14,20)(16,22)(21,24)Polycyclic presentation:
2^8|a^2=f,c^2=fg,e^2=g,f^2=h,b^a=be,d^a=dh,e^a=eg,e^b=eg76 conjugacy classes — view the full character table.
| maximal subgroup of | #136650, #136652, #136653, … |
| maximal normal subgroup of | #136650, #136652, #136653, … |
| maximal quotient of | #182492, #182529, #183455, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10890);