Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32947 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2847 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
667 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 667 conjugacy classes of subgroups — too many to draw. Only its 145 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:
(2,7)(5,8)(9,17)(10,18)(11,19)(12,25)(13,20)(15,21)(22,31)(23,28)(24,29)(27,32);(1,2,8,19,6,11,5,7)(3,9,20,28,14,23,13,17)(4,10,21,29,16,24,15,18)(12,22,30,31,25,32,26,27);(1,3)(2,9)(4,12)(5,13)(6,14)(7,17)(8,20)(10,22)(11,23)(15,26)(16,25)(18,27)(19,28)(21,30)(24,32)(29,31);(1,4)(2,10)(3,12)(5,15)(6,16)(7,18)(8,21)(9,22)(11,24)(13,26)(14,25)(17,27)(19,29)(20,30)(23,32)(28,31)Polycyclic presentation:
2^8|b^2=eh,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^c=gh,f^d=fh46 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25936);