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:
(1,2,6,10,7,11,5,12)(3,13,15,8,16,25,22,24)(4,17,19,27,21,9,20,28)(14,29,18,23,31,26,30,32);(1,3)(2,8)(4,18)(5,22)(6,15)(7,16)(9,26)(10,25)(11,24)(12,13)(14,20)(17,29)(19,31)(21,30)(23,27)(28,32);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,23)(10,27)(11,17)(12,28)(13,26)(15,30)(16,31)(18,22)(24,32)(25,29)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,e^a=eh,c^b=cf,d^b=dh,f^b=fh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11032, #359071, #359575, … |
| maximal normal subgroup of | C2×#11032, #359071, #359575, … |
| maximal quotient of | #111293, #111486, #111573, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4020);