Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9263 |
| automorphism | Aut(G) = C23×#2403 (order 1024) · Inn(G) = #386 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | #525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
239 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 239 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7,9,16)(3,11,13,30)(4,15)(10,22,21,31)(12,29,17,14)(19,28)(20,27)(23,24);(1,2,4,16,18,21,28,22)(3,8,17,23,29,32,11,27)(5,10,15,31,6,9,19,7)(12,20,14,25,30,24,13,26);(1,3)(2,8)(4,17)(5,14)(6,13)(7,20)(9,26)(10,25)(11,28)(12,19)(15,30)(16,23)(18,29)(21,32)(22,27)(24,31)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=gh,e^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=egh,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9263, #198494, #198495, … |
| maximal normal subgroup of | C2×#9263, #198494, #198495, … |
| maximal quotient of | #99915, #99916, #99919, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2251);