Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7221 |
| automorphism | Aut(G) order 4096 |
| 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 | #103 |
| 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) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14)(17,18,21,26,23,29,40,28)(19,30,32,43,35,45,34,48)(20,36,38,25,39,27,22,41)(24,33,42,31,44,47,46,37);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)(17,19,22,33,40,32,39,31,23,35,38,47,21,34,20,37)(18,24,27,43,26,46,41,30,29,44,36,48,28,42,25,45)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fgh,d^2=h,e^2=fh,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 | C2×#7224, #127276, #127277, … |
| maximal normal subgroup of | C2×#7224, #127276, #127277, … |
| maximal quotient of | #93310, #93348, #93364, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 212);