Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#537 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
666 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 666 conjugacy classes of subgroups — too many to draw. Only its 127 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:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,25)(17,22)(18,27)(19,23)(21,26)(24,28);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,19)(17,26)(18,24)(21,22)(23,25)(27,28);(2,4)(5,7)(6,10)(8,11)(9,12)(13,16,19,15)(14,21,23,28)(17,26,18,24)(20,27,25,22);(7,11)(10,12)(13,17,19,18)(14,22,23,27)(15,24,16,26)(20,28,25,21)Polycyclic presentation:
2^8|c^2=h,d^2=h,f^2=h,b^a=be,c^a=cf,f^a=fh,c^b=cg,d^b=dh,d^c=de,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #249947, #249951, #249964, … |
| maximal normal subgroup of | #249947, #249951, #249964, … |
| maximal quotient of | #327027, #327035, #327043, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22596);