Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11858 |
| automorphism | Aut(G) = #72306287 (order 4096) · Inn(G) = C2×#102 · Out(G) = C22×#41 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | #466 | C22 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
153 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 153 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,12,14)(13,16)(17,18,22,28,24,30,23,29)(19,31,34,46,36,47,35,42)(20,37,21,40,39,27,38,26)(25,41,43,33,45,32,44,48);(1,3)(2,6)(4,12)(5,11)(7,15)(8,9)(10,13)(14,16)(17,19)(18,25)(20,33)(21,41)(22,35)(23,34)(24,36)(26,31)(27,46)(28,44)(29,43)(30,45)(32,38)(37,42)(39,48)(40,47);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20,23,38,24,39,22,21)(18,26,28,37,30,40,29,27)(19,32,35,33,36,41,34,48)(25,42,43,31,45,46,44,47)Polycyclic presentation:
2^8|a^2=e,c^2=fh,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gh,c^b=cfh,d^b=dg,e^b=efgh,f^b=fh,g^b=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #131724, #131725, #131736, … |
| maximal normal subgroup of | #131724, #131725, #131736, … |
| maximal quotient of | #124300, #124302, #124456, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4854);