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) = C26×C42 |
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 | |
1332 conjugacy classes of subgroups; only the 449 normal subgroups are shown.
This group has 1332 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)(13,14,17,19)(15,16,20,18);(6,9)(7,11)(13,15,17,20)(14,18,19,16);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,15)(17,18)(19,20);(5,7)(6,10)(8,11)(9,12)(13,15)(14,16)(17,20)(18,19);(5,6)(7,10)(8,9)(11,12)(13,16)(14,15)(17,18)(19,20)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=fg,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 | C4⋊#5450 |
| maximal subgroup of | #130266, #133566, #485907, … |
| maximal normal subgroup of | #130266, #133566, #485907, … |
| maximal quotient of | #557058, #557374, #557688, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38467);