Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8534 |
| automorphism | Aut(G) = #72292028 (order 4096) · Inn(G) = #379 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | #3485 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
504 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,17,14,18,24,30)(11,15,21,27,20,26,34,38)(12,16,23,29,22,28,35,39)(19,31,33,40,32,25,36,37);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,25)(17,27)(18,26)(22,32)(23,33)(24,34)(28,31)(29,37)(30,38)(35,36)(39,40);(9,12)(10,16)(11,19)(13,23)(14,22)(15,25)(17,29)(18,28)(20,32)(21,33)(24,35)(26,31)(27,37)(30,39)(34,36)(38,40)Polycyclic presentation:
2^8|a^2=f,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^b=eh,f^b=fg,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | #133840, #135517, #139704, … |
| maximal normal subgroup of | #133840, #135517, #139704, … |
| maximal quotient of | #97052, #97053, #97055, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1527);