Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20671 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#117 |
| 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 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
646 conjugacy classes of subgroups; only the 261 normal subgroups are shown.
This group has 646 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:
(1,2,6,10,7,11,5,20)(3,12,16,23,14,27,15,8)(4,17,18,24,19,9,21,25)(13,28,30,32,29,26,31,22);(1,3,5,15,7,14,6,16)(2,8,10,12,11,23,20,27)(4,13,18,30,19,29,21,31)(9,22,24,26,17,32,25,28);(1,4)(2,9)(3,13)(5,18)(6,21)(7,19)(8,22)(10,24)(11,17)(12,26)(14,29)(15,30)(16,31)(20,25)(23,32)(27,28);(3,14)(8,23)(12,27)(13,29)(15,16)(22,32)(26,28)(30,31)Polycyclic presentation:
2^8|a^2=g,b^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #138352, #138353, #148018, … |
| maximal normal subgroup of | #138352, #138353, #148018, … |
| maximal quotient of | #188755, #188756, #188759, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13661);