Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
305 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 305 conjugacy classes of subgroups — too many to draw. Only its 123 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) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,7)(3,4,8,6)(10,13)(11,15)(17,18,21,23)(19,22,27,30)(20,24,28,32)(25,29,26,31);(1,3,5,8)(2,6,7,4)(9,10,12,13)(11,14,15,16)(17,19,21,27)(18,22,23,30)(20,25,28,26)(24,29,32,31);(9,11,12,15)(10,14,13,16)(19,25)(20,28)(22,29)(24,32)(26,27)(30,31);(1,4,5,6)(2,3,7,8)(10,13)(14,16)(18,23)(22,30)(24,32)(29,31)Polycyclic presentation:
2^8|a^2=efh,b^2=eh,c^2=f,d^2=ef,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #125809, #125994, #126459, … |
| maximal normal subgroup of | #125809, #125994, #126459, … |
| maximal quotient of | #385446, #385448, #386829, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25680);