Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
751 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 751 conjugacy classes of subgroups — too many to draw. Only its 247 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,12,16,13,15,19,22)(11,14,18,21,17,20,23,24);(2,4)(9,11,12,18,13,17,19,23)(10,14,16,21,15,20,22,24);(6,8)(11,17)(14,20)(18,23)(21,24);(10,15)(14,20)(16,22)(21,24)Polycyclic presentation:
2^8|a^2=g,b^2=g,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | #116024, #116025, #116112, … |
| maximal normal subgroup of | #116024, #116025, #116112, … |
| maximal quotient of | #191684, #191685, #191686, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13823);