Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #380 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
186 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18,20,24,21,25,27,23)(19,26,29,22,30,31,32,28);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,27,32,21,30,20,29)(18,22,24,31,25,28,23,26);(3,9)(6,12)(10,15)(13,16)(17,18)(19,28)(20,23)(21,25)(22,30)(24,27)(26,32)(29,31)Polycyclic presentation:
2^8|a^2=f,b^2=dg,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,c^b=ch,e^c=eg,f^c=fg52 conjugacy classes — view the full character table.
| maximal subgroup of | #131640, #131641, #131646, … |
| maximal normal subgroup of | #131640, #131641, #131646, … |
| maximal quotient of | #98848, #98849, #98850, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1595);