Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32894 |
| automorphism | Aut(G) = C2×#72240903 (order 12288) · Inn(G) = #2846 · Out(G) = C22×S4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2846 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | H(7,𝔽2) | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1805 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 1805 conjugacy classes of subgroups — too many to draw. Only its 181 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:
(1,2,7,13)(3,14,17,27)(4,10,19,26)(5,12,6,11)(8,20,24,16)(9,21,25,18)(15,22,32,29)(23,31,28,30);(1,3)(2,8)(4,15)(5,20)(6,16)(7,17)(9,22)(10,23)(11,27)(12,14)(13,24)(18,31)(19,32)(21,30)(25,29)(26,28);(1,4)(2,9)(3,15)(5,18)(6,21)(7,19)(8,22)(10,11)(12,26)(13,25)(14,28)(16,30)(17,32)(20,31)(23,27)(24,29);(1,4)(2,10)(3,15)(5,21)(6,18)(7,19)(8,23)(9,11)(12,25)(13,26)(14,29)(16,31)(17,32)(20,30)(22,27)(24,28)Polycyclic presentation:
2^8|a^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,e^b=eh,g^b=gh,f^c=fh,e^d=eh46 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25884);