Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18376 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
435 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 435 conjugacy classes of subgroups — too many to draw. Only its 129 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10)(3,11,14,15)(4,9,5,8)(7,12,16,13)(18,22)(20,27)(23,31)(24,28)(25,29)(30,32);(1,3,6,14)(2,7,10,16)(4,13,5,12)(8,11,9,15)(17,18,21,24)(19,23,26,32)(20,28,27,22)(25,30,29,31);(17,19,21,26)(18,23,24,32)(20,29,27,25)(22,30,28,31);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(18,24)(22,28)(23,32)(30,31)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=h,d^2=gh,e^2=g,b^a=be,c^a=cf,d^a=dgh,e^a=eg,d^b=dg,e^b=eg,f^b=fh,e^c=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | #142774, #142775, #142790, … |
| maximal normal subgroup of | #142774, #142775, #142790, … |
| maximal quotient of | #169672, #169736, #169800, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11399);