Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1703 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)(6,7)(9,10,14,17)(11,12,15,20)(13,21,22,16)(18,24,23,19);(5,6,8,7)(9,11)(10,12)(13,18)(14,15)(16,19)(17,20)(21,24)(22,23);(1,2,3,4)(5,7,8,6)(9,12)(10,15)(11,17)(13,19)(14,20)(16,23)(18,21)(22,24);(9,12)(10,15)(11,17)(13,19)(14,20)(16,23)(18,21)(22,24);(9,13,14,22)(10,16,17,21)(11,18,15,23)(12,19,20,24)Polycyclic presentation:
2^8|a^2=h,b^2=f,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 | #112017, #112031, #112052, … |
| maximal normal subgroup of | #112017, #112031, #112052, … |
| maximal quotient of | #516428, #516431, #516617, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28305);