Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9443 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
224 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 63 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9)(4,12,14,7)(5,11)(6,8)(10,13,16,15)(17,18,21,26,22,25,20,24)(19,23,27,30,28,31,29,32);(1,3,9,2,6,11,5,8)(4,10,13,7,14,16,15,12)(17,19,20,27,22,28,21,29)(18,23,26,32,25,31,24,30);(1,4)(2,7)(3,10)(5,15)(6,14)(8,12)(9,13)(11,16)(17,20,22,21)(18,24,25,26)(19,27,28,29)(23,30,31,32)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=h,d^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,f^c=fgh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132613, #132615, #132617, … |
| maximal normal subgroup of | #132613, #132615, #132617, … |
| maximal quotient of | #100129, #100130, #100145, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2442);