Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | #2388 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
456 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 149 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,24,20,22,27,29)(19,23,26,30,25,28,31,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(18,22)(19,25)(24,29)(26,31);(3,8)(5,7)(10,14)(11,15)(17,19)(18,23)(20,25)(21,26)(22,28)(24,30)(27,31)(29,32);(2,6)(3,8)(10,14)(13,16)(18,22)(19,25)(24,29)(26,31)Polycyclic presentation:
2^8|a^2=g,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cef,d^b=deh,d^c=dh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #23312⋊C3 |
| maximal subgroup of | #136278, #137271, #139976, … |
| maximal normal subgroup of | #136278, #137271, #139976, … |
| maximal quotient of | #357283, #357324, #357349, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16300);