Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#121 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#380 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
654 conjugacy classes of subgroups; only the 261 normal subgroups are shown.
This group has 654 conjugacy classes of subgroups — too many to draw. Only its 261 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:
(1,2)(3,12)(4,17,19,24)(5,20)(6,10)(7,11)(8,15)(9,18,25,21)(13,22,28,32)(14,27)(16,23)(26,29,31,30);(1,3,5,15,7,14,6,16)(2,8,10,12,11,23,20,27)(4,13,18,29,19,28,21,30)(9,22,24,26,25,32,17,31);(1,4)(2,9)(3,13)(5,18)(6,21)(7,19)(8,22)(10,24)(11,25)(12,26)(14,28)(15,29)(16,30)(17,20)(23,32)(27,31);(3,14)(8,23)(12,27)(13,28)(15,16)(22,32)(26,31)(29,30)Polycyclic presentation:
2^8|a^2=g,b^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh,f^c=fh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #360321, #360585, #360593, … |
| maximal normal subgroup of | #360321, #360585, #360593, … |
| maximal quotient of | #188781, #188785, #188789, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13663);