Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| 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 | |
401 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 401 conjugacy classes of subgroups — too many to draw. Only its 119 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:
(1,2,5,7)(3,8,4,6)(9,10,12,14)(11,15,16,13)(17,18,20,23)(19,25,27,22)(21,24,29,32)(26,31,28,30);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19,20,27)(18,22,23,25)(21,26,29,28)(24,30,32,31);(10,14)(13,15)(19,26)(21,29)(22,30)(24,32)(25,31)(27,28);(1,4,5,3)(2,6,7,8)(18,23)(22,25)(24,32)(30,31)Polycyclic presentation:
2^8|a^2=e,b^2=e,d^2=efh,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23828, #6432643, #6432651, … |
| maximal normal subgroup of | C2×#23828, #6432643, #6432651, … |
| maximal quotient of | #244474, #244480, #244486, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16816);