Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32947 |
| automorphism | Aut(G) = C2×#72240903 (order 12288) · Inn(G) = #2847 · Out(G) = C22×S4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2847 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3027 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
653 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 653 conjugacy classes of subgroups — too many to draw. Only its 145 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:
(2,7)(5,16)(8,17)(9,18)(10,19)(11,23)(12,25)(14,26)(20,30)(21,28)(22,29)(27,31);(1,2,5,7,6,10,16,19)(3,8,12,17,13,21,25,28)(4,9,14,18,15,22,26,29)(11,20,24,27,23,31,32,30);(1,3)(2,8)(4,11)(5,12)(6,13)(7,17)(9,20)(10,21)(14,24)(15,23)(16,25)(18,27)(19,28)(22,31)(26,32)(29,30);(1,4)(2,9)(3,11)(5,14)(6,15)(7,18)(8,20)(10,22)(12,24)(13,23)(16,26)(17,27)(19,29)(21,31)(25,32)(28,30)Polycyclic presentation:
2^8|b^2=e,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^c=gh,f^d=fh46 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25935);