Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
361 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 361 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,17)(8,16)(9,12)(10,23)(11,14)(13,20)(15,25)(18,34)(19,33)(21,28)(22,30)(24,32)(26,31)(27,35)(29,36);(1,2,3,4)(6,12)(13,27)(14,16)(15,28)(17,33)(23,30)(29,32)(31,34);(5,7)(6,13)(8,18)(9,19)(10,24)(11,26)(12,27)(14,29)(15,31)(16,32)(17,23)(20,25)(21,35)(22,36)(28,34)(30,33);(5,8,9,11)(6,14,12,16)(7,18,19,26)(10,21,22,25)(13,29,27,32)(15,30,28,23)(17,31,33,34)(20,24,35,36)Polycyclic presentation:
2^8|b^2=eh,d^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fg,g^a=gh,f^c=fg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #138868, #138869, #138874, … |
| maximal normal subgroup of | #138868, #138869, #138874, … |
| maximal quotient of | #490432, #490433, #490457, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26572);