Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 524288 |
| 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 | C23×#41 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2615 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)(9,10,13,14)(11,15,17,21)(12,19,20,16)(18,24,23,22);(5,6)(7,8)(10,14)(11,17)(16,19)(18,23);(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24);(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23);(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)Polycyclic presentation:
2^8|a^2=h,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 | #111953, #112289, #113657, … |
| maximal normal subgroup of | #111953, #112289, #113657, … |
| maximal quotient of | #516386, #516400, #516579, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28293);