Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | Aut(G) order 16384 |
| 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 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
585 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 585 conjugacy classes of subgroups — too many to draw. Only its 127 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)(7,12)(8,11)(9,22)(10,15)(13,19)(14,20)(16,26)(17,23)(18,32)(21,25)(24,33)(27,31)(28,30)(29,34)(35,36);(6,11)(8,19)(9,20)(12,23)(13,14)(15,25)(17,31)(18,30)(21,33)(22,24)(26,34)(27,28)(29,36)(32,35);(2,4)(5,7)(6,12)(8,17)(9,18)(10,16)(11,23)(13,27)(14,28)(15,26)(19,31)(20,30)(21,29)(22,32)(24,35)(25,34)(33,36);(7,16)(12,26)(17,29)(18,30)(23,34)(27,35)(28,32)(31,36)Polycyclic presentation:
2^8|e^2=gh,g^2=h,b^a=be,c^a=cf,e^a=eg,g^a=gh,e^b=eg,g^b=gh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #138939, #138940, #138947, … |
| maximal normal subgroup of | #138939, #138940, #138947, … |
| maximal quotient of | #484199, #484545, #484624, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26510);