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 | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
630 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 630 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:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,24)(16,25)(17,21)(18,23)(19,22)(20,27)(26,28);(6,9)(10,12)(13,15,17,25)(14,16,21,24)(18,26,19,20)(22,28,23,27);(2,4)(5,7)(6,10)(8,11)(9,12)(13,16,17,24)(14,20,21,26)(15,23,25,22)(18,28,19,27);(7,11)(10,12)(14,21)(16,24)(22,23)(27,28)Polycyclic presentation:
2^8|b^2=h,c^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gh,d^c=de,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #249980, #249984, #249995, … |
| maximal normal subgroup of | #249980, #249984, #249995, … |
| maximal quotient of | #326738, #326744, #326746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22621);