Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7734 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
161 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,6,9)(3,10,12,15)(4,8,5,13)(7,14,16,11)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,31,32,28)(33,34,37,41)(35,38,43,47)(36,39,44,40)(42,45,48,46);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,32)(30,31)(33,35,37,43)(34,38,41,47)(36,42,44,48)(39,45,40,46);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,32)(29,31)(34,39)(36,44)(38,45)(40,41)(42,48)(46,47)Polycyclic presentation:
2^8|a^2=fgh,b^2=fh,c^2=f,d^2=f,e^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,d^b=df,e^b=eg,d^c=dg,e^c=eh37 conjugacy classes — view the full character table.
| Fitting subgroup of | #7832⋊C3 |
| maximal subgroup of | C2×#7832, #165815, #165834, … |
| maximal normal subgroup of | C2×#7832, #165815, #165834, … |
| maximal quotient of | #97548, #99363, #99660, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 820);