Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#537 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
741 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 741 conjugacy classes of subgroups — too many to draw. Only its 175 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:
(6,7)(9,10)(11,15)(12,17)(13,16)(14,18)(19,22)(20,24)(21,23);(1,2)(3,4)(5,6)(7,8)(9,11)(10,15)(12,21)(13,19)(14,20)(16,22)(17,23)(18,24);(2,4)(5,7,8,6)(10,16)(12,13)(15,22)(17,18)(19,21)(23,24);(9,12,14,13)(10,17,18,16)(11,19,20,21)(15,22,24,23)Polycyclic presentation:
2^8|c^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,f^a=fh,c^b=cg,d^b=dh,d^c=dh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #394664, #394994, #395449, … |
| maximal normal subgroup of | #394664, #394994, #395449, … |
| maximal quotient of | #238718, #238722, #238726, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16476);