Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9495 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
262 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,8,5,7,9,15)(3,6,11,13,10,14,12,16)(17,18)(19,22)(20,27,28,23)(21,24)(25,29,31,32)(26,30);(1,3,9,11,5,10,4,12)(2,6,8,16,7,14,15,13)(17,19,21,26)(18,22,24,30)(20,25,28,31)(23,29,27,32);(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,31)(30,32)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dg,e^b=eh,f^b=fg,d^c=dh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132586, #132588, #132642, … |
| maximal normal subgroup of | #132586, #132588, #132642, … |
| maximal quotient of | #100199, #100200, #100217, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2497);