Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle, prime radical | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
375 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 375 conjugacy classes of subgroups — too many to draw. Only its 179 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) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,10,11,9,7,8,12)(13,14)(15,22)(16,20)(17,26)(18,21)(19,24)(23,28)(25,27);(5,7,10,12,9,6,8,11)(13,15)(14,19)(16,25)(17,23)(18,24)(20,28)(21,22)(26,27);(1,2,3,4)(5,8,9,10)(6,11,7,12)(13,16,18,17)(14,20,21,26)(15,23,24,25)(19,27,22,28);(13,17,18,16)(14,20,21,26)(15,23,24,25)(19,28,22,27)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=ef,d^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 | #122397, #128499, #134794, … |
| maximal normal subgroup of | #122397, #128499, #134794, … |
| maximal quotient of | #219272, #219273, #219274, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15228);