Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11247 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #387 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1442 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
288 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 109 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:
(1,2,6,12,7,10,5,11)(3,13,15,28,16,8,22,24)(4,17,20,27,19,9,21,26)(14,23,31,32,30,29,18,25);(1,3)(2,8)(4,18,19,31)(5,15)(6,22)(7,16)(9,25,17,32)(10,13)(11,24)(12,28)(14,20,30,21)(23,26,29,27);(1,4)(2,9,10,17)(3,14)(5,21)(6,20)(7,19)(8,23,13,29)(11,27,12,26)(15,31)(16,30)(18,22)(24,32,28,25)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=d,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,g^b=gh,e^c=eh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11278, #358643, #360369, … |
| maximal normal subgroup of | C2×#11278, #358643, #360369, … |
| maximal quotient of | #120638, #120726, #120730, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4266);