Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
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 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
417 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 417 conjugacy classes of subgroups — too many to draw. Only its 139 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:
(2,4)(5,6)(7,16)(8,13)(9,23)(10,25)(11,15)(12,19)(14,17)(18,29)(20,35)(21,31)(22,28)(24,36)(26,33)(27,34)(30,32);(5,7,17,32,10,20,24,34,11,21,22,33,8,18,9,19)(6,12,23,29,13,26,28,31,15,27,36,35,25,30,14,16);(1,2,3,4);(5,8,11,10)(6,13,15,25)(7,18,21,20)(9,22,24,17)(12,26,27,30)(14,23,28,36)(16,29,31,35)(19,33,34,32)Polycyclic presentation:
2^8|b^2=eg,c^2=f,d^2=h,e^2=gh,g^2=h,b^a=be,c^a=cf,e^a=eg,g^a=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | #137535, #141260, #141261, … |
| maximal normal subgroup of | #137535, #141260, #141261, … |
| maximal quotient of | #491432, #491449, #491465, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26701);