Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 | |
1831 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)(6,7)(12,19)(16,21)(18,23)(22,24);(5,6,8,7)(10,14)(11,17)(16,21)(18,23);(1,2,3,4)(5,7,8,6)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24);(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)Polycyclic presentation:
2^8|b^2=f,c^2=gh,e^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 | #112561, #112593, #112609, … |
| maximal normal subgroup of | #112561, #112593, #112609, … |
| maximal quotient of | #516427, #516430, #516616, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28317);