Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #102 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | C4×#102 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
214 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 214 conjugacy classes of subgroups — too many to draw. Only its 93 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:
(1,2,5,12,8,14,7,24)(3,15,18,27,21,30,20,25)(4,10,22,29,23,13,6,11)(9,19,26,16,28,32,31,17);(1,3,6,19,7,20,23,17,8,21,22,32,5,18,4,16)(2,9,13,27,12,26,11,30,14,28,10,25,24,31,29,15);(1,4,5,22,8,23,7,6)(2,10,12,29,14,13,24,11)(3,16,18,32,21,17,20,19)(9,25,26,15,28,27,31,30)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=gh,e^2=h,f^2=g,g^2=h,b^a=bd,d^a=dg,f^a=fg,g^a=gh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #128675, #128695, #128714, … |
| maximal normal subgroup of | #128675, #128695, #128714, … |
| maximal quotient of | #128537, #128592, #128605, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5634);