Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18376 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
493 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 493 conjugacy classes of subgroups — too many to draw. Only its 127 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:
(1,2,6,11)(3,13,17,29)(4,10,5,20)(7,12,23,30)(8,14,26,15)(9,19,21,22)(16,24,18,28)(25,31,27,32);(1,3,6,17)(2,8,11,26)(4,15,5,14)(7,18,23,16)(9,28,21,24)(10,29,20,13)(12,27,30,25)(19,31,22,32);(1,2)(3,8)(4,9)(5,21)(6,11)(7,12)(10,22)(13,31)(14,24)(15,28)(16,25)(17,26)(18,27)(19,20)(23,30)(29,32);(1,4,6,5)(2,9,11,21)(3,14,17,15)(7,19,23,22)(8,24,26,28)(10,12,20,30)(13,27,29,25)(16,31,18,32)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dgh,e^a=eg,d^b=dg,e^b=eg,f^b=fh,e^c=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | #245523, #245527, #245535, … |
| maximal normal subgroup of | #245523, #245527, #245535, … |
| maximal quotient of | #169620, #169632, #169684, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11371);