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) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1455 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,8)(9,10,13,16)(11,14,19,17)(12,20,21,15)(18,23,24,22);(5,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,21)(22,24);(1,2,3,4)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24);(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)Polycyclic presentation:
2^8|a^2=h,c^2=g,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 | #111714, #111728, #114052, … |
| maximal normal subgroup of | #111714, #111728, #114052, … |
| maximal quotient of | #516428, #516437, #516579, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28304);