Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #392 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1438 |
| prime radical | C2×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,10,7,11,5,20)(3,12,14,8,15,24,21,23)(4,9,18,26,19,28,17,27)(13,25,30,32,31,29,16,22);(1,3)(2,8)(4,16)(5,21)(6,14)(7,15)(9,25)(10,24)(11,23)(12,20)(13,18)(17,31)(19,30)(22,27)(26,32)(28,29);(1,4)(2,9)(3,13)(5,17)(6,18)(7,19)(8,22)(10,26)(11,28)(12,29)(14,30)(15,31)(16,21)(20,27)(23,32)(24,25)Polycyclic presentation:
2^8|a^2=g,d^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11037, #359071, #359548, … |
| maximal normal subgroup of | C2×#11037, #359071, #359548, … |
| maximal quotient of | #111355, #111356, #111357, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4025);