Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| 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 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
337 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 337 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) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,16)(8,13)(9,24)(10,22)(11,15)(12,19)(14,17)(18,29)(20,35)(21,31)(23,28)(25,36)(26,33)(27,34)(30,32);(5,7,17,32,10,20,25,34,11,21,23,33,8,18,9,19)(6,12,24,29,13,26,28,31,15,27,36,35,22,30,14,16);(2,4)(5,6)(7,12)(8,22)(9,14)(10,13)(11,15)(16,19)(17,24)(18,30)(20,26)(21,27)(23,36)(25,28)(29,32)(31,34)(33,35);(5,8,11,10)(6,13,15,22)(7,18,21,20)(9,23,25,17)(12,26,27,30)(14,24,28,36)(16,29,31,35)(19,33,34,32)Polycyclic presentation:
2^8|b^2=eg,d^2=h,e^2=gh,g^2=h,b^a=be,c^a=cf,e^a=eg,g^a=gh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #153086, #153087, #153241, … |
| maximal normal subgroup of | #153086, #153087, #153241, … |
| maximal quotient of | #490477, #490548, #490580, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26598);