Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7398 |
| automorphism | Aut(G) = #72285576 (order 2048) · Inn(G) = #1343 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1343 |
| commutator | C22×C8 | C2×C4 |
| Frattini, cosocle | #408 | C22 |
| prime radical | #2018 | C2 |
| LCS | G ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #408 ◁ G | |
| LECS | G ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
127 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 127 conjugacy classes of subgroups — too many to draw. Only its 21 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 — 2 generators on 32 points:
(1,2,4,9,6,10,18,28)(3,12,15,26,16,8,23,25)(5,11,19,27,14,24,17,31)(7,20,21,13,22,29,32,30);(2,7)(3,13)(4,17)(5,14)(8,11)(9,26)(10,22)(12,24)(16,30)(18,19)(20,29)(21,27)(25,28)(31,32)Polycyclic presentation:
2^8|a^2=d,c^2=fh,d^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=cf,d^b=def,f^b=fh,d^c=dg,e^d=eh25 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7400, #150179, #150218, … |
| maximal normal subgroup of | C2×#7400, #150179, #150218, … |
| maximal quotient of | #93207, #93211, #93223, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 388);