Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #2388 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
932 conjugacy classes of subgroups; only the 407 normal subgroups are shown.
This group has 932 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:
(3,7)(6,8)(10,13)(11,15)(17,18,20,23)(19,22,24,21);(1,2)(3,6)(4,5)(7,8)(10,13)(14,16)(17,19,20,24)(18,21,23,22);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,24)(21,23);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,23)(21,24);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|a^2=f,b^2=f,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.
| Fitting subgroup of | #52958⋊C3 |
| maximal subgroup of | #132604, #132714, #487416, … |
| maximal normal subgroup of | #132604, #132714, #487416, … |
| maximal quotient of | #582148, #582363, #582575, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45946);