Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18192 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #1595 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
497 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 497 conjugacy classes of subgroups — too many to draw. Only its 113 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 — 4 generators on 32 points:
(2,7)(3,11)(4,15)(8,21)(9,19)(10,16)(12,20)(14,25)(17,28)(22,31)(23,29)(26,30);(1,2)(3,12)(4,16)(5,10)(6,9)(7,15)(8,11)(13,20)(14,26)(17,29)(18,23)(19,28)(21,24)(22,25)(27,30)(31,32);(4,17)(7,19)(13,27)(15,28)(16,29)(20,30)(21,31)(24,32);(1,3,6,14)(2,8,9,22)(4,13,17,27)(5,11,18,25)(7,20,19,30)(10,12,23,26)(15,24,28,32)(16,21,29,31)Polycyclic presentation:
2^8|d^2=h,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,d^b=dg,e^b=eg,f^b=fh,e^c=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | #486092, #486096, #486104, … |
| maximal normal subgroup of | #486092, #486096, #486104, … |
| maximal quotient of | #167033, #167035, #167097, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11196);