Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33691 |
| automorphism | Aut(G) = #72305500 (order 2048) · Inn(G) = #3109 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #3109 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
268 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 93 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:
(3,11)(4,15)(5,19)(7,20)(9,24)(10,23)(12,17)(13,28)(14,29)(16,30)(18,31)(21,32)(22,27)(25,26);(1,2,6,8)(3,7,13,22)(4,16,18,9)(5,10,19,23)(11,27,28,20)(12,26,14,21)(15,30,31,24)(17,32,29,25);(1,3)(2,7)(4,17)(5,12)(6,13)(8,22)(9,25)(10,26)(11,15)(14,19)(16,32)(18,29)(20,24)(21,23)(27,30)(28,31);(2,8)(3,12,13,14)(4,18)(7,21,22,26)(10,23)(11,17,28,29)(15,31)(20,32,27,25)Polycyclic presentation:
2^8|b^2=h,d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^b=dh,f^b=fh,d^c=dg,e^c=eh,f^c=fg,g^c=gh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #143170, #143171, #143173, … |
| maximal normal subgroup of | #143170, #143171, #143173, … |
| maximal quotient of | #494055, #494996, #495480, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26680);