Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60388 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#120 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
2333 conjugacy classes of subgroups; only the 557 normal subgroups are shown.
This group has 2333 conjugacy classes of subgroups — too many to draw. Only its 557 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 — 5 generators on 32 points:
(1,2,7,4)(3,11,15,24)(5,9,18,16)(6,17,21,10)(8,20,12,14)(13,25,27,32)(19,29,30,23)(22,31,26,28);(1,3)(2,8)(4,12)(5,13)(6,20)(7,15)(9,22)(10,24)(11,17)(14,21)(16,26)(18,27)(19,31)(23,32)(25,29)(28,30);(1,4,7,2)(3,12,15,8)(5,16,18,9)(6,10,21,17)(11,20,24,14)(13,26,27,22)(19,23,30,29)(25,31,32,28);(1,5)(2,9)(3,13)(4,16)(6,19)(7,18)(8,22)(10,23)(11,25)(12,26)(14,28)(15,27)(17,29)(20,31)(21,30)(24,32);(5,18)(9,16)(13,27)(19,30)(22,26)(23,29)(25,32)(28,31)Polycyclic presentation:
2^8|a^2=h,c^2=h,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gh,f^c=fh,e^d=eh58 conjugacy classes — view the full character table.
| Fitting subgroup of | #60391⋊C3, #72252447 |
| maximal subgroup of | #152028, #499884, #499889, … |
| maximal normal subgroup of | #152028, #499884, #499889, … |
| maximal quotient of | #6350445, #6350446, #6350447, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53379);