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×#525 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1533 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)(9,10,14,12)(11,15,17,19)(13,16,21,20)(18,22,23,24);(5,6)(7,8)(11,17)(13,21)(15,19)(16,20);(1,2,3,4)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23);(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)Polycyclic presentation:
2^8|a^2=h,c^2=g,d^2=h,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 | #97456, #129459, #133864, … |
| maximal normal subgroup of | #97456, #129459, #133864, … |
| maximal quotient of | #515289, #515293, #515329, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28621);