Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2069 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #396 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1443 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,13)(4,17,21,9)(5,11)(6,12)(7,10)(8,15)(14,25,29,30)(16,28)(18,23,31,32)(19,26,20,27)(22,24);(1,3,7,15,6,16,5,22)(2,8,10,28,12,24,11,13)(4,18,19,14,21,31,20,29)(9,25,27,32,17,30,26,23);(1,4,5,19,6,21,7,20)(2,9,10,26,12,17,11,27)(3,14,15,18,16,29,22,31)(8,23,13,25,24,32,28,30)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=d,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=dh,e^b=eh,e^c=eh,g^c=gh46 conjugacy classes — view the full character table.
| Fitting subgroup of | #11311⋊C3 |
| maximal subgroup of | C2×#11311, #359387, #365327, … |
| maximal normal subgroup of | C2×#11311, #359387, #365327, … |
| maximal quotient of | #121067, #121099, #121102, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4299);