Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,21,24,22,25,20,29)(19,26,28,23,27,32,30,31);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(19,27)(23,31)(26,32)(28,30);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19)(18,23)(20,30)(21,28)(22,27)(24,32)(25,31)(26,29);(2,8)(6,9)(7,16)(14,15)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ceh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #107811, #107814, #107974, … |
| maximal normal subgroup of | #107811, #107814, #107974, … |
| maximal quotient of | #215661, #215662, #215663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14833);