Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29592 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
351 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 351 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10,12,13)(11,15,16,14)(19,25)(22,31)(26,32)(27,30);(2,5)(6,8)(10,13)(14,15)(17,18,20,23)(19,26,25,32)(21,29,28,24)(22,30,31,27);(3,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,19,20,25)(18,22,23,31)(21,30,28,27)(24,32,29,26);(1,3,4,7)(2,6,5,8)(11,16)(14,15)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=h,d^2=efh,g^2=h,b^a=be,c^a=cf,c^b=cg,g^b=gh,d^c=de,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #166891, #166893, #166923, … |
| maximal normal subgroup of | #166891, #166893, #166923, … |
| maximal quotient of | #380415, #380471, #381286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23120);