Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
368 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 368 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,14)(13,16)(17,18,20,23)(19,25,27,22)(21,24,29,32)(26,31,28,30);(2,5)(3,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,29)(31,32);(10,14)(13,15)(19,26)(21,29)(22,30)(24,32)(25,31)(27,28);(1,3,4,8)(2,6,5,7)(11,16)(13,15)(18,23)(19,27)(24,32)(26,28)Polycyclic presentation:
2^8|a^2=ef,d^2=efh,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #123059, #123069, #6647790, … |
| maximal normal subgroup of | #123059, #123069, #6647790, … |
| maximal quotient of | #384814, #385822, #386648, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25533);