Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30236 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2644 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,20,7,9,5,11)(3,12,15,23,16,21,13,8)(4,10,18,25,19,24,17,26)(14,27,29,32,30,31,28,22);(1,3)(2,8,9,23)(4,14)(5,15)(6,13)(7,16)(10,22,24,32)(11,12,20,21)(17,29)(18,28)(19,30)(25,31,26,27);(2,9)(3,13)(4,17)(5,6)(8,21)(10,25)(12,23)(15,16)(18,19)(24,26)(27,31)(28,29);(1,4,6,18,7,19,5,17)(2,10,20,25,9,24,11,26)(3,14,13,28,16,30,15,29)(8,22,21,31,23,32,12,27)Polycyclic presentation:
2^8|a^2=fg,b^2=f,d^2=eh,e^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gh,c^b=cg,g^b=gh,d^c=de,e^c=eh,g^c=gh34 conjugacy classes — view the full character table.
| Fitting subgroup of | #30237⋊C3 |
| maximal subgroup of | #507894, #507897, #508805, … |
| maximal normal subgroup of | #507894, #507897, #508805, … |
| maximal quotient of | #380190, #380395, #382343, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23225);