Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29610 |
| 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 | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
470 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 470 conjugacy classes of subgroups — too many to draw. Only its 121 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:
(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(17,18)(19,25)(20,28)(21,24)(22,26)(23,29)(27,32)(30,31);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(20,29)(22,31)(23,28)(26,30);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,19,21,27)(18,22,24,31)(20,30,29,26)(23,32,28,25);(4,12)(7,14)(10,16)(13,15)Polycyclic presentation:
2^8|a^2=e,b^2=e,c^2=h,f^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gh,c^b=cg,f^b=fh,d^c=de,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #248818, #248820, #248825, … |
| maximal normal subgroup of | #248818, #248820, #248825, … |
| maximal quotient of | #327774, #327775, #327784, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22640);