Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19651 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3169 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
356 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 356 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) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(4,7)(8,12)(9,13)(11,14)(15,16)(17,18)(19,26)(20,24)(21,30)(22,25)(23,28)(27,31)(29,32);(1,2,7,13,5,9,4,6)(3,8,14,16,10,15,11,12);(1,3)(2,8)(4,11)(5,10)(6,12)(7,14)(9,15)(13,16)(17,19,21,28,22,29,20,27)(18,23,24,26,25,31,30,32);(3,10)(8,15)(11,14)(12,16)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,26,31,32)Polycyclic presentation:
2^8|b^2=eg,c^2=f,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^c=dg52 conjugacy classes — view the full character table.
| maximal subgroup of | #141369, #141370, #141824, … |
| maximal normal subgroup of | #141369, #141370, #141824, … |
| maximal quotient of | #191517, #206143, #206144, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12654);