Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | Aut(G) = #72339816 (order 8192) · Inn(G) = #555 · Out(G) = #1600 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #555 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2537 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
410 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 410 conjugacy classes of subgroups — too many to draw. Only its 125 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)(3,13)(4,18)(5,8)(6,11)(7,12)(9,16)(10,20)(14,23)(15,24)(17,25)(19,22)(21,26)(27,28)(29,31)(30,32);(2,8,12,11)(3,14,17,16)(4,19)(9,23)(10,18,26,22)(13,25)(15,29,30,28)(20,21);(1,3)(2,9)(4,15)(5,14)(6,16)(7,17)(8,13)(10,24)(11,25)(12,23)(18,31)(19,30)(20,28)(21,29)(22,27)(26,32);(1,4,5,20,7,19,6,21)(2,10,11,18,12,26,8,22)(3,15,14,28,17,30,16,29)(9,24,25,31,23,32,13,27)Polycyclic presentation:
2^8|b^2=eg,d^2=e,e^2=h,f^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,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #151280, #151281, #509105, … |
| maximal normal subgroup of | #151280, #151281, #509105, … |
| maximal quotient of | #317904, #317905, #317906, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21238);