Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #397 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1440 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
402 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 109 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,10,7,12,5,11)(3,13,15,8,16,24,21,25)(4,17,19,27,20,26,22,9)(14,23,30,28,18,31,29,32);(1,3)(2,8)(4,18)(5,21)(6,15)(7,16)(9,23)(10,24)(11,13)(12,25)(14,20)(17,28)(19,29)(22,30)(26,32)(27,31);(1,4,5,19,7,20,6,22)(2,9,10,26,12,27,11,17)(3,14,15,29,16,18,21,30)(8,23,13,28,25,31,24,32)Polycyclic presentation:
2^8|a^2=g,c^2=d,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11159, #359281, #360288, … |
| maximal normal subgroup of | C2×#11159, #359281, #360288, … |
| maximal quotient of | #118670, #118671, #118712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4147);