Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33758 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
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 | |
267 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 267 conjugacy classes of subgroups — too many to draw. Only its 99 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(10,14)(11,17)(12,19)(15,22)(16,23)(18,21)(20,24);(1,2)(3,6)(4,5)(7,8)(10,15)(14,22)(16,21)(18,23);(2,5)(6,8)(9,10)(11,18)(12,21)(13,15)(14,17)(16,19)(20,23)(22,24);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,c^b=ceh,f^c=fg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #489561, #489564, #489571, … |
| maximal normal subgroup of | #489561, #489564, #489571, … |
| maximal quotient of | #496590, #496591, #496701, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26805);