Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C26 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1631 conjugacy classes of subgroups; only the 463 normal subgroups are shown.
This group has 1631 conjugacy classes of subgroups — too many to draw. Only its 463 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) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(6,8)(10,13)(11,15)(17,18,21,23)(19,20,24,22);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,24)(22,23);(2,5)(6,8)(11,15)(14,16)(17,19,21,24)(18,22,23,20);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,20)(21,24)(22,23);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,19)(21,22)(23,24)Polycyclic presentation:
2^8|a^2=g,c^2=g,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #37475⋊C3, #72247529 |
| maximal subgroup of | #122191, #131443, #238082, … |
| maximal normal subgroup of | #122191, #131443, #238082, … |
| maximal quotient of | #556408, #556618, #556820, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30463);