Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
1272 conjugacy classes of subgroups; only the 449 normal subgroups are shown.
This group has 1272 conjugacy classes of subgroups — too many to draw. Only its 449 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(7,11)(10,12)(13,14)(15,19)(16,17)(18,20);(6,8)(7,11)(14,16)(18,19);(1,2,3,4)(5,6,9,8)(7,10,11,12)(13,14,17,16)(15,18,20,19);(5,7,9,11)(6,10,8,12)(13,15)(14,18)(16,19)(17,20);(5,8,9,6)(7,12,11,10)(13,16,17,14)(15,19,20,18)Polycyclic presentation:
2^8|c^2=fgh,d^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #46328⋊C3 |
| maximal subgroup of | #130229, #133545, #485918, … |
| maximal normal subgroup of | #130229, #133545, #485918, … |
| maximal quotient of | #557086, #557391, #557758, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39316);