Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #331 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C2 | C2×#331 |
| prime radical | #3147 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9,18,10,28,26,6,11,20,27,4,8,17,24)(3,12,14,29,13,7,19,22,16,21,31,25,15,23,30,32);(1,3)(2,7)(4,13)(5,19)(6,16)(8,21)(9,25)(10,12)(11,23)(14,28)(15,18)(17,31)(20,30)(22,26)(24,32)(27,29);(1,4,6,18)(2,8,11,10)(3,13,16,15)(5,17,20,28)(7,21,23,12)(9,24,27,26)(14,19,31,30)(22,25,32,29)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=g,e^2=fgh,f^2=h,b^a=bd,d^a=df,f^a=fh,g^a=gh,d^b=dg,e^b=efgh,f^b=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #94778, #94780, #128897, … |
| maximal normal subgroup of | #94778, #94780, #128897, … |
| maximal quotient of | #122896, #122898, #122899, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4910);