Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(6,9)(10,12)(13,14)(15,21)(16,24)(17,18)(19,23)(20,25)(22,26)(27,28);(1,3,4,2)(14,18)(19,21)(20,24)(26,27);(5,6)(7,10)(8,9)(11,12)(13,15,17,23)(14,19,18,21)(16,22,25,28)(20,26,24,27);(13,14)(15,19)(16,20)(17,18)(21,23)(22,26)(24,25)(27,28);(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=f,e^2=fg,b^a=bf,c^a=cg,e^a=ef,d^b=df88 conjugacy classes — view the full character table.
| maximal subgroup of | #107874, #108368, #190338, … |
| maximal normal subgroup of | #107874, #108368, #190338, … |
| maximal quotient of | #513566, #513584, #513602, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27222);