Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11663 |
| 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 | C2×#102 |
| 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 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
215 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 215 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)(7,15,16,31)(8,11)(9,21)(10,24)(12,17,26,34)(13,23)(14,20)(18,29,25,33)(19,32)(22,30,36,28)(27,35);(1,2,3,4)(6,11)(12,26)(13,27)(14,24)(15,31)(21,32)(28,30)(29,33);(5,7)(6,12)(8,16)(9,22)(10,25)(11,26)(13,29)(14,30)(15,32)(17,35)(18,20)(19,36)(21,31)(23,34)(24,28)(27,33)Polycyclic presentation:
2^8|a^2=f,b^2=dh,e^2=g,g^2=h,b^a=bd,c^a=ce,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 | #131857, #131858, #131859, … |
| maximal normal subgroup of | #131857, #131858, #131859, … |
| maximal quotient of | #134364, #134365, #134409, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5805);