Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9980 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C23×C4 | C23 |
| prime radical | #2815 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
301 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 301 conjugacy classes of subgroups — too many to draw. Only its 49 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 — 3 generators on 32 points:
(2,7)(3,12)(4,16)(8,24)(9,13)(10,21)(11,23)(14,20)(15,31)(17,18)(22,26)(25,28)(27,30)(29,32);(1,2,6,11)(3,13,14,30)(4,8,18,26)(5,9,19,27)(7,20,23,12)(10,16,28,17)(15,22,31,24)(21,29,25,32);(2,8)(3,14)(4,17)(7,21)(9,10)(11,26)(12,20)(13,24)(15,32)(16,18)(22,30)(23,25)(27,28)(29,31)Polycyclic presentation:
2^8|b^2=h,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,f^b=fg,d^c=dh,f^c=fg28 conjugacy classes — view the full character table.
| maximal subgroup of | #151344, C2×#9983, #232512, … |
| maximal normal subgroup of | #151344, C2×#9983, #232512, … |
| maximal quotient of | #102475, #102487, #102489, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2971);