Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) order 8192 |
| 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 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
359 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 359 conjugacy classes of subgroups — too many to draw. Only its 121 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)(9,21)(10,25)(11,15)(12,18)(14,23)(17,32)(19,33)(20,31)(22,27)(24,35)(26,30)(28,36)(29,34);(1,2,3,4);(5,7)(6,12)(8,17)(9,22)(10,26)(11,19)(13,27)(14,30)(15,29)(16,21)(18,23)(20,34)(24,36)(25,28)(31,32)(33,35);(5,8,11,10)(6,13,15,25)(7,17,19,26)(9,20,24,23)(12,27,29,28)(14,21,31,35)(16,32,33,30)(18,22,34,36)Polycyclic presentation:
2^8|b^2=e,d^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,f^c=fg,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #136892, #136897, #136925, … |
| maximal normal subgroup of | #136892, #136897, #136925, … |
| maximal quotient of | #490427, #490452, #490493, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26567);