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) = C210 |
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 | |
2545 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)(13,21)(17,24)(19,20)(22,23);(5,6)(7,8)(9,10,14,18)(11,16,12,15)(13,17,21,24)(19,23,20,22);(1,2)(3,4)(5,6)(7,8)(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23);(9,12,14,11)(10,16,18,15)(13,20,21,19)(17,23,24,22);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)Polycyclic presentation:
2^8|b^2=h,c^2=h,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 | #111947, #113039, #113642, … |
| maximal normal subgroup of | #111947, #113039, #113642, … |
| maximal quotient of | #516388, #516399, #516580, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28230);