Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7213 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #342 |
| commutator | C22×C4 | C42 |
| Frattini, cosocle | C2×C4×Q8 | C22 |
| prime radical | C23 | #95 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,9,6,10,5,8)(3,11,12,7,14,15,13,16)(17,18,20,24,21,25,29,32)(19,23,26,31,27,22,28,30);(1,3,5,13,6,14,4,12)(2,7,9,15,10,16,8,11)(18,22,25,23)(24,31,32,30)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fg,d^2=fg,e^2=fgh,b^a=bc,c^a=cf,e^a=efg,f^a=fh,g^a=gh,d^b=dg,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | #93826, #93827, #93830, … |
| maximal normal subgroup of | #93826, #93827, #93830, … |
| maximal quotient of | #93336, #93342, #93630, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 236);