Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11989 |
| automorphism | Aut(G) = #72306289 (order 1024) · Inn(G) = #1551 · Out(G) = C23 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1551 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| prime radical | #540 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
140 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 49 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 — 3 generators on 32 points:
(1,2,4,10,17,7,16,20,6,11,19,26,5,9,18,25)(3,12,14,27,29,23,30,31,13,24,28,32,15,8,21,22);(2,7)(4,16)(5,17)(8,12)(9,11)(10,26)(14,30)(15,29)(18,19)(21,28)(22,31)(23,24);(1,3)(2,8,11,23)(4,14)(5,15)(6,13)(7,12,9,24)(10,22,26,31)(16,30)(17,29)(18,21)(19,28)(20,27,25,32)Polycyclic presentation:
2^8|a^2=e,c^2=g,d^2=g,e^2=fgh,f^2=h,b^a=bd,c^a=cf,d^a=df,f^a=fh,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,d^c=dh34 conjugacy classes — view the full character table.
| maximal subgroup of | #132067, #132068, #132072, … |
| maximal normal subgroup of | #132067, #132068, #132072, … |
| maximal quotient of | #124350, #124351, #124410, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4979);