Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
804 conjugacy classes of subgroups; only the 505 normal subgroups are shown.
This group has 804 conjugacy classes of subgroups — too many to draw. Only its 505 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) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,7)(6,8)(9,10,13,16)(11,15,12,14);(1,3)(2,6)(4,7)(5,8)(17,18,20,22)(19,23,24,21);(9,11,13,12)(10,14,16,15)(17,19,20,24)(18,21,22,23);(2,7)(3,8)(9,12,13,11)(10,15,16,14);(1,4,5,6)(2,3,7,8)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=gh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df64 conjugacy classes — view the full character table.
| Fitting subgroup of | #52287⋊C3, #72266670 |
| maximal subgroup of | #98768, #129624, #131151, … |
| maximal normal subgroup of | #98768, #129624, #131151, … |
| maximal quotient of | #520812, #520820, #521093, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45275);