Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18346 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
512 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 512 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,5)(4,16)(9,25)(10,23)(11,20)(12,24)(14,29)(15,21)(17,32)(18,31)(19,28)(22,27)(26,30);(1,2)(3,8)(4,17)(5,20)(6,11)(7,12)(9,19)(10,13)(14,30)(15,24)(16,26)(18,22)(21,23)(25,29)(27,28)(31,32);(1,3,13,21)(2,8,10,23)(4,14,28,29)(5,7,15,6)(9,22,26,32)(11,20,12,24)(16,31,19,18)(17,30,27,25);(1,4,13,28)(2,9,10,26)(3,14,21,29)(5,18,15,31)(6,19,7,16)(8,22,23,32)(11,17,12,27)(20,30,24,25)Polycyclic presentation:
2^8|c^2=gh,d^2=gh,e^2=g,b^a=be,c^a=cf,d^a=dh,e^a=eg,d^b=dg,e^b=eg,f^b=fh,e^c=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #245905, #245909, #245932, … |
| maximal normal subgroup of | #245905, #245909, #245932, … |
| maximal quotient of | #168955, #168956, #169011, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11358);