Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #476 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
207 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 207 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,17)(8,21,23,28)(9,25)(10,12)(11,16)(13,19,31,35)(14,18)(15,20)(22,34,24,33)(26,30,36,32)(27,29);(1,2,3,4)(5,7,18,27,11,20,9,10)(6,12,25,15,16,29,14,17)(8,19,24,36,23,35,22,26)(13,28,32,33,31,21,30,34);(5,8)(6,13)(7,19)(9,22)(10,26)(11,23)(12,28)(14,30)(15,33)(16,31)(17,34)(18,24)(20,35)(21,29)(25,32)(27,36)Polycyclic presentation:
2^8|a^2=f,b^2=dh,d^2=h,e^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,e^c=egh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #131842, #131846, #134972, … |
| maximal normal subgroup of | #131842, #131846, #134972, … |
| maximal quotient of | #134369, #134414, #134445, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5812);