Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | Aut(G) = #72338260 (order 4096) · Inn(G) = #555 · Out(G) = #392 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #555 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2563 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
304 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 117 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:
(1,2,6,12)(3,9)(4,16,17,22)(5,8,7,11)(10,19,26,18)(13,21)(14,28)(15,23)(20,25)(24,27)(29,32)(30,31);(2,8,12,11)(3,13,15,20)(4,17)(9,23)(10,16,26,22)(14,29,30,27)(18,19)(21,25);(1,3)(2,9)(4,14)(5,13)(6,15)(7,20)(8,21)(10,24)(11,25)(12,23)(16,28)(17,30)(18,27)(19,29)(22,31)(26,32);(1,4,5,18,6,17,7,19)(2,10,11,16,12,26,8,22)(3,14,13,27,15,30,20,29)(9,24,25,28,23,32,21,31)Polycyclic presentation:
2^8|a^2=efg,b^2=eg,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #510081, #510082, #510151, … |
| maximal normal subgroup of | #510081, #510082, #510151, … |
| maximal quotient of | #326502, #326503, #326504, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22143);