Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | Aut(G) = C22×#511819 (order 2048) · Inn(G) = #535 · Out(G) = #120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #535 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2598 |
| prime radical | #2010 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
331 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 331 conjugacy classes of subgroups — too many to draw. Only its 101 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)(3,12)(4,8)(5,9)(6,11)(7,14)(10,18)(13,27)(15,20)(16,24)(17,25)(19,32)(21,30)(22,29)(23,31)(26,28);(1,3,5,14,6,15,18,30)(2,7,10,12,11,21,9,20)(4,13,17,29,16,28,31,32)(8,22,23,27,24,19,25,26);(1,4)(2,8)(3,13)(5,17)(6,16)(7,19)(9,25)(10,23)(11,24)(12,26)(14,29)(15,28)(18,31)(20,27)(21,22)(30,32);(2,9,11,10)(4,16)(7,20,21,12)(8,23,24,25)(13,28)(17,31)(19,26,22,27)(29,32)Polycyclic presentation:
2^8|b^2=e,d^2=g,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gh,c^b=cg,f^b=fh,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | #485588, #485589, C2×#29385, … |
| maximal normal subgroup of | #485588, #485589, C2×#29385, … |
| maximal quotient of | #326615, #326616, #326617, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22373);