Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | #3002 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1037 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,16)(11,20,12,18)(13,19,15,23)(17,22,24,21);(2,5)(6,8)(10,16)(17,24)(18,20)(19,23);(9,11,14,12)(10,17,16,24)(13,21,15,22)(18,23,20,19);(1,3,4,7)(2,6,5,8)(9,12,14,11)(10,18,16,20)(13,21,15,22)(17,23,24,19);(9,13,14,15)(10,19,16,23)(11,21,12,22)(17,18,24,20)Polycyclic presentation:
2^8|a^2=h,c^2=h,d^2=f,e^2=h,g^2=h,b^a=bf,c^a=cg,g^a=gh,g^c=gh,e^d=eh61 conjugacy classes — view the full character table.
| maximal subgroup of | #249897, #249901, #249905, … |
| maximal normal subgroup of | #249897, #249901, #249905, … |
| maximal quotient of | #6371019, #6371820, #6371828, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53577);