Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) = C2×#72237167 (order 4096) · Inn(G) = C2×D8 · Out(G) = C2×#525 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C2×#102 |
| prime radical | #466 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
155 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 155 conjugacy classes of subgroups — too many to draw. Only its 59 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:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,28)(20,25)(21,33)(22,37)(23,27)(24,30)(26,35)(29,44)(31,45)(32,43)(34,39)(36,47)(38,42)(40,48)(41,46);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,24)(20,29)(21,34)(22,38)(23,31)(25,39)(26,42)(27,41)(28,33)(30,35)(32,46)(36,48)(37,40)(43,44)(45,47);(3,9)(6,12)(10,15)(13,16)(17,20,23,22)(18,25,27,37)(19,29,31,38)(21,32,36,35)(24,39,41,40)(26,33,43,47)(28,44,45,42)(30,34,46,48)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=fh,e^2=g,f^2=h,b^a=bd,d^a=df,f^a=fh,c^b=cg,d^b=df,f^b=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #131638, #131639, #131640, … |
| maximal normal subgroup of | #131638, #131639, #131640, … |
| maximal quotient of | #125077, #125079, #125123, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4635);