Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | Aut(G) order 1572864 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 | |
2414 conjugacy classes of subgroups; only the 557 normal subgroups are shown.
This group has 2414 conjugacy classes of subgroups — too many to draw. Only its 557 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 12 points:
(2,3)(10,11);(1,2,4,3)(6,8)(10,11);(9,10,12,11);(5,6,7,8)(9,11,12,10);(1,3,4,2)Polycyclic presentation:
2^8|b^2=f,c^2=g,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #59816⋊C3, #72231896, #72231915, … |
| maximal subgroup of | #97496, #98690, #130226, … |
| maximal normal subgroup of | #97496, #98690, #130226, … |
| maximal quotient of | #531806, #532488, #533373, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52804);