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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1770 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 1770 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:
(2,8)(3,12)(5,6)(10,25)(11,17)(13,27)(14,19)(15,18)(20,23)(21,26)(22,31)(28,29);(1,2)(3,9)(4,10)(5,17)(6,8)(7,11)(12,23)(13,22)(14,24)(15,21)(16,26)(18,25)(19,20)(27,32)(28,30)(29,31);(1,3,7,14)(2,9,11,24)(4,13,16,29)(5,12,6,19)(8,20,17,23)(10,22,26,31)(15,28,18,27)(21,30,25,32);(1,4)(2,10)(3,13)(5,18)(6,15)(7,16)(8,21)(9,22)(11,26)(12,27)(14,29)(17,25)(19,28)(20,30)(23,32)(24,31)Polycyclic presentation:
2^8|c^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, 25883);