Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11912 |
| automorphism | Aut(G) = #72306289 (order 1024) · Inn(G) = #1530 · Out(G) = C23 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1530 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | #97 | C23 |
| prime radical | #540 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G | |
| LECS | G ▷ #97 ▷ C2×C4 ▷ C2 ▷ C1 | |
142 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 47 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:
(1,2,4,10,5,7,19,21,6,11,18,27,20,9,16,26)(3,12,14,29,15,24,31,28,13,25,17,32,30,8,22,23);(2,7)(4,16)(5,20)(8,12)(9,11)(10,27)(14,22)(15,30)(17,31)(18,19)(23,28)(24,25);(1,3)(2,8,11,24)(4,17)(5,15)(6,13)(7,12,9,25)(10,28,27,23)(14,18)(16,31)(19,22)(20,30)(21,32,26,29)Polycyclic presentation:
2^8|a^2=e,c^2=g,d^2=g,e^2=f,f^2=h,b^a=bd,c^a=cf,d^a=df,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,d^c=dh,e^c=eh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #132067, #132068, #132072, … |
| maximal normal subgroup of | #132067, #132068, #132072, … |
| maximal quotient of | #124337, #124338, #124460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4900);