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 | #3508 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,8,5,6,11,13)(3,7,10,14,9,12,15,16)(17,18)(19,23)(20,22)(21,25)(24,28)(26,30)(27,29)(31,32);(2,6)(4,11)(7,12)(10,15)(18,22)(20,28)(23,29)(24,25)(27,32)(30,31);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,19,21,26)(18,23,25,30)(20,27,28,32)(22,29,24,31);(3,9)(7,12)(10,15)(14,16)(19,26)(23,30)(27,32)(29,31)Polycyclic presentation:
2^8|a^2=f,c^2=h,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 | #221234, #221237, #221241, … |
| maximal normal subgroup of | #221234, #221237, #221241, … |
| maximal quotient of | #160512, #160520, #160523, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10712);