Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 131072 |
| 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 | C2×#532 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1197 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,7)(8,11,12,10)(13,14,18,20)(15,21,22,16)(17,25,26,19)(23,24,28,27);(6,7)(10,11)(15,22)(16,21)(23,28)(24,27);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,16)(17,23)(18,22)(19,24)(20,21)(25,27)(26,28);(5,8)(6,10)(7,11)(9,12)(13,16)(14,15)(17,24)(18,21)(19,23)(20,22)(25,28)(26,27);(13,17,18,26)(14,19,20,25)(15,23,22,28)(16,24,21,27)Polycyclic presentation:
2^8|a^2=h,c^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #147832, #147834, #498907, … |
| maximal normal subgroup of | #147832, #147834, #498907, … |
| maximal quotient of | #553723, #553985, #554179, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30364);