Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29742 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
385 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18,20,26,22,25,21,24)(19,23,27,30,28,32,29,31);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,20,27,22,28,21,29)(18,23,26,30,25,32,24,31);(3,7)(6,8)(10,14)(13,16)(18,24)(19,27)(20,21)(25,26)(28,29)(30,31);(1,3,4,7)(2,6,5,8)(18,25)(19,28)(24,26)(27,29)Polycyclic presentation:
2^8|a^2=egh,b^2=f,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=dh,d^c=de,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #295172, #295174, #295175, … |
| maximal normal subgroup of | #295172, #295174, #295175, … |
| maximal quotient of | #380170, #380171, #380609, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23213);