Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
833 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 833 conjugacy classes of subgroups — too many to draw. Only its 199 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)(13,14)(15,17)(16,18)(19,20)(21,22,24,26)(23,27,28,25);(6,9)(10,12)(14,15)(16,20)(21,22,24,26)(23,25,28,27);(5,6,8,9)(7,10,11,12)(13,15,17,14)(16,19,20,18)(21,22)(23,25)(24,26)(27,28);(5,7,8,11)(6,10,9,12)(13,16,17,20)(14,18,15,19)(21,23)(22,25)(24,28)(26,27)Polycyclic presentation:
2^8|a^2=h,b^2=ef,c^2=efg,d^2=efg,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de64 conjugacy classes — view the full character table.
| maximal subgroup of | #139816, C2×#14516, #520908, … |
| maximal normal subgroup of | #139816, C2×#14516, #520908, … |
| maximal quotient of | #153962, #153970, #154021, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7504);