Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C22×C4⋊C4 |
| prime radical | C24 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
319 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 319 conjugacy classes of subgroups — too many to draw. Only its 187 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,11)(12,16)(13,15)(17,18,20,24,21,23,27,30)(19,22,26,29,25,28,31,32);(1,3,5,11,6,13,4,12)(2,7,8,10,9,16,14,15)(17,19)(18,22)(20,26)(21,25)(23,28)(24,29)(27,31)(30,32);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,16,10)(19,25)(22,28)(26,31)(29,32);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,10,16,15)(18,23)(22,28)(24,30)(29,32)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=g,d^2=g,e^2=g,f^2=h,b^a=be,d^a=dh,e^a=eg,c^b=cgh64 conjugacy classes — view the full character table.
| maximal subgroup of | #116492, #116493, #116500, … |
| maximal normal subgroup of | #116492, #116493, #116500, … |
| maximal quotient of | #180931, #183397, #183411, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10807);