Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
453 conjugacy classes of subgroups; only the 195 normal subgroups are shown.
This group has 453 conjugacy classes of subgroups — too many to draw. Only its 195 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)(6,9)(10,12)(13,14)(15,19)(16,23)(17,26)(18,21)(20,24)(22,28)(25,27);(5,6)(7,10)(8,9)(11,12)(13,15,18,17)(14,19,21,26)(16,24,25,22)(20,27,28,23);(5,7,8,11)(6,10,9,12)(13,16,17,24,18,25,15,22)(14,20,19,23,21,28,26,27);(6,9)(10,12)(16,25)(20,28)(22,24)(23,27)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=ef,f^2=h,b^a=be,c^a=cf,f^a=fh,c^b=ch,d^b=de,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #138460, #138461, C2×#22714, … |
| maximal normal subgroup of | #138460, #138461, C2×#22714, … |
| maximal quotient of | #226217, #226227, #226237, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15702);