Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
912 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 912 conjugacy classes of subgroups — too many to draw. Only its 151 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:
(1,2)(3,4)(9,10)(11,16)(12,21)(13,17)(14,22)(15,18)(19,23)(20,24);(2,4)(5,6)(7,8)(9,11,15,20)(10,16,18,24)(12,14,19,13)(17,21,22,23);(6,8)(9,12)(10,16)(11,13)(14,20)(15,19)(17,23)(18,24)(21,22);(2,4)(9,13,15,14)(10,17,18,22)(11,19,20,12)(16,23,24,21)Polycyclic presentation:
2^8|b^2=h,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #123013, #123023, C2×#23929, … |
| maximal normal subgroup of | #123013, #123023, C2×#23929, … |
| maximal quotient of | #245123, #245124, #245125, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16917);