Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | Aut(G) order 1572864 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1905 conjugacy classes of subgroups; only the 567 normal subgroups are shown.
This group has 1905 conjugacy classes of subgroups — too many to draw. Only its 567 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 — 5 generators on 24 points:
(2,4)(6,7)(9,10,14,18)(11,15,19,24)(12,16,22,21)(13,17,23,20);(5,6,8,7)(11,19)(13,23)(15,24)(17,20);(1,2,3,4)(5,7,8,6)(9,11)(10,15)(12,17)(13,21)(14,19)(16,23)(18,24)(20,22);(9,12)(10,16)(11,20)(13,15)(14,22)(17,19)(18,21)(23,24);(9,13)(10,17)(11,21)(12,15)(14,23)(16,19)(18,20)(22,24)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=g,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28451);