Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #386 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C24 | C24 |
| socle | C2 | C2×#386 |
| prime radical | #3449 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
874 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)(3,7,13,20)(4,14,16,8)(5,9)(6,10)(11,18)(12,19,25,30)(15,27,28,21)(17,22)(23,29)(24,26)(31,32);(1,3)(2,7,10,20)(4,11)(5,12)(6,13)(8,18,14,26)(9,19,22,30)(15,23)(16,24)(17,25)(21,29,27,32)(28,31);(1,4)(2,8)(3,11)(5,15)(6,16)(7,18)(9,21)(10,14)(12,23)(13,24)(17,28)(19,29)(20,26)(22,27)(25,31)(30,32);(1,5,6,17)(2,9,10,22)(3,12,13,25)(4,15,16,28)(7,19,20,30)(8,21,14,27)(11,23,24,31)(18,29,26,32)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=eh,f^b=fh,g^b=gh,e^c=eh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #146610, #146615, #147795, … |
| maximal normal subgroup of | #146610, #146615, #147795, … |
| maximal quotient of | #196310, #196312, #196313, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14002);