Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
635 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 635 conjugacy classes of subgroups — too many to draw. Only its 223 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,9,10,8,12,7,11)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28);(5,7,8,9)(6,10,12,11)(13,15,17,23)(14,18,20,21)(16,24,25,22)(19,27,28,26);(2,4)(5,7,8,9)(6,11,12,10)(13,16,17,25)(14,19,20,28)(15,22,23,24)(18,26,21,27);(6,12)(10,11)(14,20)(18,21)(19,28)(26,27)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=ceh76 conjugacy classes — view the full character table.
| maximal subgroup of | #106339, #106342, #106531, … |
| maximal normal subgroup of | #106339, #106342, #106531, … |
| maximal quotient of | #215724, #215725, #215726, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14808);