Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23414 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#127 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#392 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
990 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 990 conjugacy classes of subgroups — too many to draw. Only its 181 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:
(1,2)(3,11)(4,8)(5,17)(6,10)(7,13)(9,19)(12,27)(14,25)(15,23)(16,31)(18,21)(20,32)(22,29)(24,30)(26,28);(1,3)(2,7)(4,12)(5,18)(6,14)(8,22)(9,25)(10,21)(11,17)(13,19)(15,28)(16,32)(20,23)(24,26)(27,31)(29,30);(1,4)(2,8)(3,12)(5,16)(6,15)(7,20)(9,24)(10,23)(11,26)(13,29)(14,28)(17,31)(18,32)(19,30)(21,22)(25,27);(4,15)(8,23)(12,28)(16,30)(20,22)(24,31)(26,27)(29,32)Polycyclic presentation:
2^8|e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gh,c^b=cg,e^b=eh,f^b=fh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #142260, #142261, #393928, … |
| maximal normal subgroup of | #142260, #142261, #393928, … |
| maximal quotient of | #232706, #232707, #232708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16402);