Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
769 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 769 conjugacy classes of subgroups — too many to draw. Only its 163 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:
(2,4)(7,11)(10,12)(13,14)(15,17)(16,19)(18,20)(22,25)(26,27);(6,9)(10,12)(14,15)(16,20)(21,22,24,25)(23,27,28,26);(5,6,8,9)(7,10,11,12)(13,15,17,14)(16,19,20,18)(21,23,24,28)(22,26,25,27);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,16)(14,18)(15,19)(17,20)(21,22)(23,26)(24,25)(27,28)Polycyclic presentation:
2^8|b^2=ef,c^2=g,d^2=eg,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de58 conjugacy classes — view the full character table.
| maximal subgroup of | #142557, C2×#15941, #609802, … |
| maximal normal subgroup of | #142557, C2×#15941, #609802, … |
| maximal quotient of | #154809, #154915, #154972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8929);