Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17603 |
| 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 | C2×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(18,22)(20,27)(23,29)(24,28)(26,31)(30,32);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,18,21,24)(19,23,25,30)(20,28,27,22)(26,32,31,29);(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,31)(28,32);(19,25)(23,30)(26,31)(29,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,e^2=h,f^2=gh,b^a=be,e^a=eg,e^b=eh,f^b=fgh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #220658, #220659, #220661, … |
| maximal normal subgroup of | #220658, #220659, #220661, … |
| maximal quotient of | #163742, #163755, #163758, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10762);