Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | Aut(G) order 4096 |
| 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 | |
459 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,8,5,6,11,13)(3,7,10,14,9,12,15,16)(17,18)(19,26)(20,23)(21,25)(22,30)(24,28)(27,31)(29,32);(2,6)(4,11)(7,12)(10,15)(17,19)(18,24)(20,27)(21,28)(22,31)(23,29)(25,32)(26,30);(1,3,5,9)(2,7,6,12)(4,10,11,15)(8,14,13,16)(17,20,23,18)(19,27,29,24)(21,25,22,30)(26,28,32,31);(3,9)(7,12)(10,15)(14,16)(17,21,23,22)(18,25,20,30)(19,28,29,31)(24,32,27,26)Polycyclic presentation:
2^8|a^2=f,c^2=g,d^2=h,e^2=h,f^2=gh,b^a=be,c^a=ch,e^a=eg,e^b=eh,f^b=fgh,d^c=dg49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#17768, #6344072, #6344073, … |
| maximal normal subgroup of | C2×#17768, #6344072, #6344073, … |
| maximal quotient of | #160656, #160661, #160662, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10756);