Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19621 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
386 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 386 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,13,15)(11,17,18,24)(12,20,21,16)(14,22,23,19);(9,11,12,19,13,18,21,22)(10,14,16,17,15,23,20,24);(1,2,3,5,4,6,7,8)(10,15)(14,23)(16,20)(17,24);(11,18)(14,23)(17,24)(19,22)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=fg,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dg52 conjugacy classes — view the full character table.
| maximal subgroup of | #289172, #289179, #289280, … |
| maximal normal subgroup of | #289172, #289179, #289280, … |
| maximal quotient of | #191433, #206108, #206111, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12612);