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 | #3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
646 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)(3,12)(4,16,19,27)(5,20)(6,10)(7,11)(8,14)(9,18,26,17)(13,25,30,22)(15,23)(21,24)(28,29,32,31);(1,3)(2,8)(4,13)(5,21)(6,14)(7,15)(9,25)(10,23)(11,24)(12,20)(16,32)(17,31)(18,29)(19,30)(22,26)(27,28);(1,4)(2,9)(3,13)(5,18)(6,17)(7,19)(8,22)(10,27)(11,26)(12,28)(14,31)(15,30)(16,20)(21,29)(23,32)(24,25)Polycyclic presentation:
2^8|a^2=g,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=dh,d^c=dh,e^c=eh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11003, #359071, #359649, … |
| maximal normal subgroup of | C2×#11003, #359071, #359649, … |
| maximal quotient of | #111321, #111325, #111563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3991);