Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#124 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#394 |
| prime radical | C2×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
668 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 668 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,15)(5,19)(6,10)(7,13)(8,17)(9,21)(12,26)(14,25)(16,30)(18,24)(20,23)(22,27)(28,31)(29,32);(1,3)(2,7)(4,16)(5,20)(6,14)(8,22)(9,25)(10,23)(11,19)(12,17)(13,21)(15,28)(18,27)(24,30)(26,29)(31,32);(1,4,5,18,6,17,21,32)(2,8,9,24,10,15,19,29)(3,12,13,27,14,16,20,31)(7,22,11,26,23,28,25,30);(4,17)(8,15)(12,16)(18,32)(22,28)(24,29)(26,30)(27,31)Polycyclic presentation:
2^8|c^2=e,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gh,c^b=cg,e^b=eh,f^b=fh,d^c=dh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #142304, #142305, #394598, … |
| maximal normal subgroup of | #142304, #142305, #394598, … |
| maximal quotient of | #238875, #238876, #238877, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16515);