Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
401 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 401 conjugacy classes of subgroups — too many to draw. Only its 115 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18,23,29,22,20,21,25)(19,24,30,32,26,28,27,31);(1,3,4,8)(2,5,6,7)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,28)(21,27)(22,26)(23,30)(25,31)(29,32);(11,15)(13,16)(17,20,22,18)(19,24,26,28)(21,25,23,29)(27,32,30,31);(2,6)(5,7)(19,26)(24,28)(27,30)(31,32)Polycyclic presentation:
2^8|a^2=fg,b^2=egh,c^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #142577, #142579, #142581, … |
| maximal normal subgroup of | #142577, #142579, #142581, … |
| maximal quotient of | #361743, #364359, #365142, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22106);