Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | Aut(G) order 4096 |
| 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 | C2×C4×D4 |
| prime radical | #3100 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #431 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
323 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 323 conjugacy classes of subgroups — too many to draw. Only its 145 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)(3,7)(4,6)(5,8)(9,12)(10,15)(11,13)(14,16)(17,18,20,24,21,23,27,30)(19,22,26,29,25,28,31,32);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(19,25)(22,28)(26,31)(29,32);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(17,19,20,26,21,25,27,31)(18,22,24,29,23,28,30,32);(2,8)(3,10)(6,9)(11,16)(18,23)(22,28)(24,30)(29,32)Polycyclic presentation:
2^8|a^2=f,c^2=f,e^2=g,f^2=h,b^a=be,d^a=dgh,e^a=eg,c^b=ch,e^b=eg,d^c=dg52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#17950, #6344725, #6344732, … |
| maximal normal subgroup of | C2×#17950, #6344725, #6344732, … |
| maximal quotient of | #182523, #182559, #183634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10938);