Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9495 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator, prime radical | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
239 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 239 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) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,6,10,13)(4,7)(5,8)(9,12,15,16)(11,14)(17,18,21,25,22,19,23,27)(20,26,29,32,28,24,30,31);(1,3,5,10)(2,6)(4,9,11,15)(7,12)(8,13)(14,16)(17,19,21,27,22,18,23,25)(20,26,30,31,28,24,29,32);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,12,13,16)(17,20)(18,24)(19,26)(21,30)(22,28)(23,29)(25,32)(27,31)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=g,f^2=h,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 | #132822, #132824, #132826, … |
| maximal normal subgroup of | #132822, #132824, #132826, … |
| maximal quotient of | #100192, #100194, #100196, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2502);