Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11855 |
| 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 | |
155 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 155 conjugacy classes of subgroups — too many to draw. Only its 57 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)(17,18,22,28,24,29,23,39)(19,30,33,47,34,45,40,41)(20,27,21,37,36,46,35,26)(25,38,42,48,44,31,43,32);(1,3)(2,6)(4,12)(5,11)(7,15)(8,9)(10,13)(14,16)(17,19)(18,25)(20,32)(21,38)(22,33)(23,40)(24,34)(26,45)(27,41)(28,42)(29,44)(30,37)(31,35)(36,48)(39,43)(46,47);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20,23,35,24,36,22,21)(18,26,28,27,29,37,39,46)(19,31,33,32,34,38,40,48)(25,41,43,45,44,47,42,30)Polycyclic presentation:
2^8|a^2=e,c^2=f,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=cf,d^b=dg,e^b=efgh,f^b=fh,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131712, #131713, #131736, … |
| maximal normal subgroup of | #131712, #131713, #131736, … |
| maximal quotient of | #124296, #124298, #124453, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4849);