Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33595 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#459 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
353 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 353 conjugacy classes of subgroups — too many to draw. Only its 117 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 — 4 generators on 32 points:
(2,7)(3,11)(4,15)(5,16)(8,22)(9,20)(10,26)(12,17)(13,28)(14,29)(18,30)(19,24)(23,27)(25,32);(1,2,4,7,6,9,15,20)(3,8,12,19,14,23,13,25)(5,10,16,21,18,26,30,31)(11,24,28,22,29,32,17,27);(1,3)(2,8)(4,12)(5,17)(6,14)(7,19)(9,23)(10,27)(11,16)(13,15)(18,28)(20,25)(21,24)(22,26)(29,30)(31,32);(2,9)(7,20)(8,23)(10,26)(19,25)(21,31)(22,27)(24,32)Polycyclic presentation:
2^8|b^2=e,e^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fg,g^a=gh,d^b=dh,f^c=fg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #500646, #500648, #500650, … |
| maximal normal subgroup of | #500646, #500648, #500650, … |
| maximal quotient of | #490434, #490436, #490439, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26583);