Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17603 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
435 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 435 conjugacy classes of subgroups — too many to draw. Only its 179 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:
(1,2,5,10,6,9,4,8)(3,11,12,16,13,7,14,15)(18,22)(20,27)(23,29)(24,28)(26,31)(30,32);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,18,21,24)(19,23,25,30)(20,28,27,22)(26,32,31,29);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(17,19,21,25)(18,23,24,30)(20,26,27,31)(22,29,28,32);(19,25)(23,30)(26,31)(29,32)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=g,e^2=h,f^2=gh,b^a=be,e^a=eg,e^b=eh,f^b=fgh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #152683, C2×#17780, #6344148, … |
| maximal normal subgroup of | #152683, C2×#17780, #6344148, … |
| maximal quotient of | #163749, #163750, #163753, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10768);