Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7221 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #340 |
| commutator | C2×C8 | C42 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| prime radical | C23 | #106 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
92 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10)(11,20,22,32)(12,25)(13,28)(14,19)(15,21,30,23)(16,26)(17,33)(18,29)(24,39,36,35)(27,34)(31,38,40,37);(2,5)(3,7)(9,11,12,23,13,24,26,38,14,22,27,21,29,36,33,37)(10,15,17,20,18,31,34,35,19,30,16,32,28,40,25,39)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fgh,d^2=gh,e^2=fgh,f^2=h,b^a=bc,c^a=cf,e^a=efg,f^a=fh,c^b=ch,d^b=dg40 conjugacy classes — view the full character table.
| maximal subgroup of | #93846, #93852, C2×#7246, … |
| maximal normal subgroup of | #93846, #93852, C2×#7246, … |
| maximal quotient of | #93332, #93348, #93635, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 234);