Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19697 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
427 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 427 conjugacy classes of subgroups — too many to draw. Only its 123 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)(9,10,15,19)(11,20,23,24)(12,17,16,22)(13,18,14,21);(2,6)(5,8)(9,11,14,17,15,23,13,22)(10,16,18,20,19,12,21,24);(1,2,3,5,4,6,7,8)(9,12)(10,17)(11,21)(13,20)(14,24)(15,16)(18,23)(19,22);(9,13,15,14)(10,18,19,21)(11,22,23,17)(12,24,16,20)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=f,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,c^b=ch,d^c=dg40 conjugacy classes — view the full character table.
| maximal subgroup of | #219574, #219599, #219616, … |
| maximal normal subgroup of | #219574, #219599, #219616, … |
| maximal quotient of | #191454, #206201, #206206, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12689);