Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | #3050 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
562 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 562 conjugacy classes of subgroups — too many to draw. Only its 131 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:
(1,2)(3,4)(5,6)(7,15)(8,19)(9,24)(10,12)(11,16)(13,22)(14,21)(17,35)(18,32)(20,30)(23,31)(25,29)(26,34)(27,36)(28,33);(5,7,10,18)(6,11,12,26)(8,20,23,36)(9,25,21,17)(13,29,28,35)(14,30,24,27)(15,31,32,19)(16,33,34,22);(2,4);(6,12)(11,26)(13,28)(14,24)(15,32)(19,31)(27,30)(29,35)Polycyclic presentation:
2^8|b^2=h,e^2=gh,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,g^a=gh,e^b=eg,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #137385, #137389, #484198, … |
| maximal normal subgroup of | #137385, #137389, #484198, … |
| maximal quotient of | #484195, #484541, #484620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26506);