Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
325 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 325 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,15)(13,16)(14,22)(18,21)(19,24)(20,23);(2,5)(6,8)(10,15)(16,23)(17,22)(18,24);(10,16)(11,14)(13,21)(15,23)(17,18)(19,20)(22,24);(1,3,4,7)(2,6,5,8)(9,11,12,14)(10,17,15,22)(13,19,20,21)(16,24,23,18)Polycyclic presentation:
2^8|d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^c=dh,f^c=fg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #137031, #137039, #137473, … |
| maximal normal subgroup of | #137031, #137039, #137473, … |
| maximal quotient of | #494029, #494968, #495465, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26657);