Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1029 conjugacy classes of subgroups; only the 407 normal subgroups are shown.
This group has 1029 conjugacy classes of subgroups — too many to draw. Only its 407 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 — 5 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(10,12)(11,15)(19,23)(21,24);(9,10,13,12)(11,16,15,14)(17,18)(19,21)(20,22)(23,24);(2,3)(6,7)(11,15)(14,16)(17,18,20,22)(19,21,23,24);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,16)(13,15)(17,18)(19,21)(20,22)(23,24);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,16,15,14)(17,19,20,23)(18,21,22,24)Polycyclic presentation:
2^8|b^2=f,c^2=h,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | #7223459, #7223463, #7223489, … |
| maximal normal subgroup of | #7223459, #7223463, #7223489, … |
| maximal quotient of | #582268, #582446, #582756, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46105);