Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C22 | C22×D8 |
| prime radical | C22×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
585 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 585 conjugacy classes of subgroups — too many to draw. Only its 223 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)(7,16)(8,14)(9,21)(10,25)(11,13)(12,18)(15,23)(17,33)(19,32)(20,30)(22,28)(24,35)(26,31)(27,34)(29,36);(5,7)(6,12)(8,17)(9,22)(10,26)(11,19)(13,27)(14,28)(15,31)(16,21)(18,23)(20,34)(24,36)(25,29)(30,33)(32,35);(6,13)(12,27)(14,25)(15,30)(16,32)(21,35)(28,29)(31,33);(5,8,11,10)(6,14,13,25)(7,17,19,26)(9,20,24,23)(12,28,27,29)(15,21,30,35)(16,33,32,31)(18,22,34,36)Polycyclic presentation:
2^8|a^2=f,d^2=h,e^2=gh,g^2=h,b^a=be,c^a=ch,e^a=eg,g^a=gh,e^b=eg,g^b=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #146730, #146731, #146747, … |
| maximal normal subgroup of | #146730, #146731, #146747, … |
| maximal quotient of | #481056, #481263, #481266, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26368);