Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1201 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(6,10)(7,12)(14,18)(19,21)(20,25)(26,27);(5,6,9,10)(7,11,12,8)(14,18)(16,24)(19,21)(22,28);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28);(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27);(5,8,9,11)(6,7,10,12)(13,16)(14,20)(15,22)(17,24)(18,25)(19,26)(21,27)(23,28)Polycyclic presentation:
2^8|a^2=fgh,b^2=f,d^2=h,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #213780, #213800, #213809, … |
| maximal normal subgroup of | #213780, #213800, #213809, … |
| maximal quotient of | #515387, #515435, #515594, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28612);