Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2657 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,8)(12,19)(16,21)(18,23)(22,24);(5,6)(7,8)(10,14)(11,17)(16,21)(18,23);(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24);(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24);(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)Polycyclic presentation:
2^8|d^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #113044, #113634, #133845, … |
| maximal normal subgroup of | #113044, #113634, #133845, … |
| maximal quotient of | #516387, #516398, #516580, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28137);