Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| 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 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
313 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,7)(3,4,8,6)(10,13)(11,15)(17,18)(19,22)(20,24)(21,23)(25,29)(26,31)(27,30)(28,32);(1,3,5,8)(2,6,7,4)(9,10,12,13)(11,14,15,16)(17,19)(18,22)(20,25)(21,27)(23,30)(24,29)(26,28)(31,32);(9,11,12,15)(10,14,13,16)(19,25)(20,28)(22,29)(24,32)(26,27)(30,31);(1,4,5,6)(2,3,7,8)(10,13)(14,16)(18,23)(22,30)(24,32)(29,31)Polycyclic presentation:
2^8|a^2=ef,b^2=e,c^2=f,d^2=ef,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #6649095, #6649098, #6649114, … |
| maximal normal subgroup of | #6649095, #6649098, #6649114, … |
| maximal quotient of | #385444, #385445, #385446, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25677);