Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13490 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2073 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
146 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 146 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:
(1,2,7,19,4,8,6,11)(3,9,13,25,14,21,18,29)(5,10,20,26,15,22,16,28)(12,24,17,27,30,23,31,32);(2,8)(3,12)(5,16)(6,7)(9,23)(10,26)(13,31)(14,30)(15,20)(17,18)(21,24)(22,28)(25,27)(29,32);(1,3)(2,9,8,21)(4,14)(5,17)(6,18)(7,13)(10,27,22,32)(11,29,19,25)(12,16)(15,31)(20,30)(23,28,24,26)Polycyclic presentation:
2^8|a^2=eg,c^2=d,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,f^b=fg,g^b=gh,e^c=eh,f^c=fg,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13490, #499369, #499603, … |
| maximal normal subgroup of | C2×#13490, #499369, #499603, … |
| maximal quotient of | #146048, #146055, #146068, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6478);