Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9938 |
| automorphism | Aut(G) = C22×#72231761 (order 4096) · Inn(G) = #392 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
287 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 287 conjugacy classes of subgroups — too many to draw. Only its 51 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:
(1,2,6,9)(3,10,12,16)(4,8,5,13)(7,14,15,11)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,31)(30,32);(1,3)(2,7)(4,11)(5,14)(6,12)(8,16)(9,15)(10,13)(17,19,21,26)(18,22,24,29)(20,25,27,32)(23,30,31,28);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,10,15,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,31)(26,32)(29,30)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=g,d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,d^c=dh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9945, #234928, #234929, … |
| maximal normal subgroup of | C2×#9945, #234928, #234929, … |
| maximal quotient of | #102988, #102991, #102993, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2933);