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) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
399 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 399 conjugacy classes of subgroups — too many to draw. Only its 127 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:
(1,2,5,8)(3,7,4,6)(9,10,12,14)(11,13,15,16)(17,18,20,24,21,25,28,23)(19,22,27,31,26,30,32,29);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,19,21,26)(18,22,25,30)(20,27,28,32)(23,29,24,31);(11,15)(13,16)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(1,4,5,3)(2,7,8,6)(19,26)(22,30)(27,32)(29,31)Polycyclic presentation:
2^8|a^2=efg,b^2=egh,d^2=e,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 | C2×#29152, #6468041, #6468049, … |
| maximal normal subgroup of | C2×#29152, #6468041, #6468049, … |
| maximal quotient of | #326537, #326538, #326545, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22140);