Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2032 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #393 |
| commutator, Frattini, cosocle | C24 | C24 |
| socle | C2 | C2×#393 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
582 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 582 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,14,17,8)(5,9)(6,10)(11,24,15,19)(12,20)(13,21)(16,27,29,22)(18,23)(25,32,28,30)(26,31);(1,3)(2,7)(4,15,17,11)(5,12)(6,13)(8,19,14,24)(9,20)(10,21)(16,28,29,25)(18,26)(22,30,27,32)(23,31);(1,4)(2,8)(3,11)(5,16)(6,17)(7,19)(9,22)(10,14)(12,25)(13,15)(18,29)(20,30)(21,24)(23,27)(26,28)(31,32);(1,5,6,18)(2,9,10,23)(3,12,13,26)(4,16,17,29)(7,20,21,31)(8,22,14,27)(11,25,15,28)(19,30,24,32)Polycyclic presentation:
2^8|a^2=e,b^2=e,d^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gh,c^b=cg,g^b=gh,e^c=eh46 conjugacy classes — view the full character table.
| Fitting subgroup of | #23474⋊C3 |
| maximal subgroup of | #359177, #375944, #391001, … |
| maximal normal subgroup of | #359177, #375944, #391001, … |
| maximal quotient of | #238072, #238075, #238076, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16462);