Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #476 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1802 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
200 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 77 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:
(1,2,5,13,8,14,7,11)(3,15,17,9,18,12,6,23)(4,19,21,30,22,32,20,28)(10,25,26,16,27,24,29,31);(1,3,5,17,8,18,7,6)(2,9,11,15,14,23,13,12)(4,16,20,25,22,31,21,24)(10,19,26,30,27,32,29,28);(1,4)(2,10)(3,16)(5,20)(6,24)(7,21)(8,22)(9,19)(11,26)(12,28)(13,29)(14,27)(15,30)(17,25)(18,31)(23,32)Polycyclic presentation:
2^8|a^2=f,b^2=d,d^2=h,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,e^c=egh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #131839, #131840, #131843, … |
| maximal normal subgroup of | #131839, #131840, #131843, … |
| maximal quotient of | #134368, #134369, #134413, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5813);