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 | |
2349 conjugacy classes of subgroups; only the 557 normal subgroups are shown.
This group has 2349 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:
(2,7)(5,16)(8,21)(9,20)(10,17)(12,18)(15,27)(19,23)(22,31)(24,28)(26,29)(30,32);(1,2)(3,8)(4,9)(5,17)(6,10)(7,16)(11,22)(12,19)(13,23)(14,24)(15,28)(18,21)(20,27)(25,32)(26,30)(29,31);(1,3,6,13)(2,8,10,23)(4,11,14,25)(5,18,16,12)(7,19,17,21)(9,22,24,32)(15,29,27,26)(20,30,28,31);(1,4)(2,9)(3,11)(5,15)(6,14)(7,20)(8,22)(10,24)(12,26)(13,25)(16,27)(17,28)(18,29)(19,30)(21,31)(23,32);(4,14)(9,24)(11,25)(15,27)(20,28)(22,32)(26,29)(30,31)Polycyclic presentation:
2^8|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 | #60390⋊C3 |
| maximal subgroup of | #152026, #499879, #499880, … |
| maximal normal subgroup of | #152026, #499879, #499880, … |
| maximal quotient of | #6350429, #6350430, #6350431, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53378);