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) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#534 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1625 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)(9,10,14,16)(11,17,18,13)(12,15,22,24)(19,23,20,21);(5,6,7,8)(11,18)(13,17)(19,20)(21,23);(1,2)(3,4)(9,11)(10,13)(12,20)(14,18)(15,23)(16,17)(19,22)(21,24);(9,12)(10,15)(11,19)(13,21)(14,22)(16,24)(17,23)(18,20);(9,13)(10,11)(12,21)(14,17)(15,19)(16,18)(20,24)(22,23)Polycyclic presentation:
2^8|a^2=h,b^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 | #97557, #133862, #133923, … |
| maximal normal subgroup of | #97557, #133862, #133923, … |
| maximal quotient of | #515286, #515291, #515327, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28435);