Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #394 |
| 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 | |
632 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)(3,11,14,7)(4,15,17,8)(5,19)(6,9)(10,22)(12,23)(13,24,20,27)(16,32)(18,26,21,28)(25,31)(29,30);(1,3)(2,7)(4,16)(5,20)(6,14)(8,25)(9,11)(10,27)(12,18)(13,22)(15,32)(17,31)(19,24)(21,30)(23,26)(28,29);(1,4)(2,8)(3,12)(5,21)(6,17)(7,23)(9,15)(10,28)(11,29)(13,31)(14,30)(16,20)(18,22)(19,26)(24,32)(25,27)Polycyclic presentation:
2^8|a^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,f^b=fh,g^b=gh,d^c=dh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11033, #359161, #359543, … |
| maximal normal subgroup of | C2×#11033, #359161, #359543, … |
| maximal quotient of | #111294, #111405, #111575, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4021);