Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
442 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 442 conjugacy classes of subgroups — too many to draw. Only its 131 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:
(2,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,24)(16,21)(17,27)(18,23)(19,20)(22,25)(26,28);(1,2,4,3)(6,9)(10,12)(13,15,18,19)(14,20,23,24)(16,26,17,25)(21,22,27,28);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,16,18,17)(14,21,23,27)(15,25,19,26)(20,28,24,22);(13,17,18,16)(14,22,23,28)(15,25,19,26)(20,27,24,21)Polycyclic presentation:
2^8|b^2=fh,c^2=e,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,g^a=gh,c^b=ch,g^d=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #273220, #273222, #273235, … |
| maximal normal subgroup of | #273220, #273222, #273235, … |
| maximal quotient of | #351111, #351113, #351115, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26027);