Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18376 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
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 | |
492 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 492 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)(4,16)(8,22)(9,20)(10,17)(12,26)(13,29)(14,15)(18,31)(21,28)(23,27)(24,32)(25,30);(1,2)(3,12)(4,17)(5,10)(6,9)(7,16)(8,14)(11,23)(13,21)(15,28)(18,32)(19,24)(20,31)(22,25)(26,29)(27,30);(4,18)(7,20)(13,29)(16,31)(17,32)(21,26)(22,27)(25,30);(1,3,6,15)(2,8,9,23)(4,13,18,29)(5,14,19,11)(7,21,20,26)(10,12,24,28)(16,30,31,25)(17,27,32,22)Polycyclic presentation:
2^8|d^2=h,e^2=g,b^a=be,c^a=cf,d^a=dgh,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 | #486098, #486102, #486110, … |
| maximal normal subgroup of | #486098, #486102, #486110, … |
| maximal quotient of | #169637, #169638, #169701, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11374);