Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1828 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #381 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C24 | C24 |
| socle | C2 | C2×#381 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
638 conjugacy classes of subgroups; only the 221 normal subgroups are shown.
This group has 638 conjugacy classes of subgroups — too many to draw. Only its 221 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)(3,11,14,7)(4,15)(5,9)(6,10)(8,17)(12,19,26,23)(13,24,27,20)(16,28)(18,22)(21,29)(25,30,32,31);(1,3)(2,7)(4,12)(5,13)(6,14)(8,19)(9,20)(10,11)(15,23)(16,25)(17,26)(18,27)(21,30)(22,24)(28,31)(29,32);(1,4)(2,8,10,15)(3,12)(5,16)(6,17)(7,19,11,23)(9,21,22,28)(13,25)(14,26)(18,29)(20,30,24,31)(27,32);(1,5,6,18)(2,9,10,22)(3,13,14,27)(4,16,17,29)(7,20,11,24)(8,21,15,28)(12,25,26,32)(19,30,23,31)Polycyclic presentation:
2^8|a^2=g,c^2=e,d^2=h,b^a=be,c^a=cf,e^a=eh,g^b=gh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #146498, #146502, #147613, … |
| maximal normal subgroup of | #146498, #146502, #147613, … |
| maximal quotient of | #190012, #190014, #190015, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13698);