Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×C22⋊C4 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C23 | C25 |
| socle | C2 | C23×C22⋊C4 |
| prime radical | C2×#585 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C23 ▷ C2 ▷ C1 | |
2446 conjugacy classes of subgroups; only the 989 normal subgroups are shown.
This group has 2446 conjugacy classes of subgroups — too many to draw. Only its 989 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)(3,11,14,7)(4,8)(5,9)(6,10)(12,23,25,18)(13,24,27,19)(15,20)(16,21)(17,22)(26,31,32,29)(28,30);(1,3)(2,7)(4,12)(5,13)(6,14)(8,18)(9,19)(10,11)(15,25)(16,26)(17,27)(20,23)(21,29)(22,24)(28,32)(30,31);(1,4)(2,8)(3,12)(5,16)(6,15)(7,18)(9,21)(10,20)(11,23)(13,26)(14,25)(17,28)(19,29)(22,30)(24,31)(27,32);(4,15)(8,20)(12,25)(16,28)(18,23)(21,30)(26,32)(29,31);(1,5,6,17)(2,9,10,22)(3,13,14,27)(4,16,15,28)(7,19,11,24)(8,21,20,30)(12,26,25,32)(18,29,23,31)Polycyclic presentation:
2^8|a^2=g,e^2=h,b^a=bf,f^a=fh,g^b=gh,d^c=dh82 conjugacy classes — view the full character table.
| Fitting subgroup of | #60070⋊C3 |
| maximal subgroup of | #146590, #355925, #355934, … |
| maximal normal subgroup of | #146590, #355925, #355934, … |
| maximal quotient of | #6342543, #6342546, #6342547, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53058);