Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
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 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
424 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 424 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,21)(22,24)(23,27)(26,31)(28,32)(29,30);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,27)(18,23,20,25)(22,31,30,28)(24,32,29,26);(1,3)(2,6)(4,8)(5,7)(10,14)(13,16)(17,20,21,18)(19,26,27,32)(22,24,30,29)(23,31,25,28);(3,8)(5,6)(11,15)(13,16)(18,20)(23,25)(24,29)(26,32)Polycyclic presentation:
2^8|b^2=h,c^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,g^b=gh,d^c=de,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #123032, #123042, C2×#32714, … |
| maximal normal subgroup of | #123032, #123042, C2×#32714, … |
| maximal quotient of | #391198, #391199, #391241, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25702);