Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17547 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| prime radical | #3511 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,9)(5,10)(6,15)(7,11)(8,13)(14,16)(17,18,21,25,22,26,20,24)(19,27,28,23,30,31,29,32);(1,3,7,4)(2,8,11,16)(5,13,6,14)(9,15,12,10)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,27)(26,32);(1,4,7,3)(2,9,11,12)(5,13,6,14)(8,15,16,10)(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,31,32,27);(1,5,7,6)(2,10,11,15)(3,13,4,14)(8,9,16,12)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=h,d^2=g,e^2=h,f^2=gh,b^a=be,e^a=eg,e^b=eh,f^b=fgh,d^c=dg61 conjugacy classes — view the full character table.
| maximal subgroup of | #221227, #221230, #221238, … |
| maximal normal subgroup of | #221227, #221230, #221238, … |
| maximal quotient of | #160515, #160516, #160519, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10715);