Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| socle | C22 | C22×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
345 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 345 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:
(1,2)(3,4)(5,6)(7,16)(8,13)(9,23)(10,25)(11,15)(12,20)(14,17)(18,29)(19,30)(21,35)(22,28)(24,36)(26,33)(27,34)(31,32);(5,7,17,32,10,21,24,33,11,18,22,34,8,19,9,20)(6,12,23,30,13,27,28,29,15,26,36,35,25,31,14,16);(2,4)(7,18)(12,26)(16,29)(19,21)(20,33)(27,31)(30,35)(32,34);(5,8,11,10)(6,13,15,25)(7,19,18,21)(9,22,24,17)(12,27,26,31)(14,23,28,36)(16,30,29,35)(20,34,33,32)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,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | #150254, #150257, #152510, … |
| maximal normal subgroup of | #150254, #150257, #152510, … |
| maximal quotient of | #490471, #490542, #490571, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26592);