Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,6)(3,7,8,5)(17,18,21,27,22,26,31,25)(19,28,30,24,20,23,29,32);(1,3,4,8)(2,5,6,7)(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,22)(21,30)(24,25)(26,28)(27,32)(29,31);(10,13)(14,16)(17,20,22,19)(18,24,26,32)(21,30,31,29)(23,27,28,25);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,22,31)(18,25,26,27)(19,29,20,30)(23,32,28,24)Polycyclic presentation:
2^8|a^2=efh,b^2=eg,c^2=h,d^2=g,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#29589, #6461951, #6461957, … |
| maximal normal subgroup of | C2×#29589, #6461951, #6461957, … |
| maximal quotient of | #363995, #364284, #371375, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22577);