Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| 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 | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,28)(21,23)(22,25)(26,32)(27,30)(29,31);(1,3,5,8)(2,6,7,4)(9,11)(10,13)(12,16)(14,15)(18,23)(19,26)(20,22)(24,31)(27,29)(30,32);(10,14)(13,15)(17,19,21,27)(18,24,23,30)(20,26,22,29)(25,31,28,32);(1,4,5,6)(2,3,7,8)(11,16)(13,15)(17,20,21,22)(18,25,23,28)(19,26,27,29)(24,31,30,32)Polycyclic presentation:
2^8|a^2=efh,b^2=efh,c^2=h,d^2=ef,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #6647789, #6647792, #6647797, … |
| maximal normal subgroup of | #6647789, #6647792, #6647797, … |
| maximal quotient of | #385071, #385072, #385487, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25645);