Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #386 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#386 |
| prime radical | C2×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
832 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)(3,13)(4,17,20,26)(5,10)(6,11)(7,12)(8,15)(9,18,25,19)(14,28,31,22)(16,23)(21,24)(27,30,32,29);(1,3,5,15,7,16,6,21)(2,8,11,13,12,24,10,23)(4,14,18,29,20,31,19,30)(9,22,26,27,25,28,17,32);(1,4)(2,9)(3,14)(5,18)(6,19)(7,20)(8,22)(10,17)(11,26)(12,25)(13,27)(15,29)(16,31)(21,30)(23,32)(24,28);(1,5,7,6)(2,10,12,11)(3,15,16,21)(4,18,20,19)(8,23,24,13)(9,17,25,26)(14,29,31,30)(22,32,28,27)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^b=fh,e^c=eh,f^c=fh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #354322, #355822, #356645, … |
| maximal normal subgroup of | #354322, #355822, #356645, … |
| maximal quotient of | #196078, #196081, #196082, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13983);