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 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1042 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)(9,10,14,13)(11,18,15,16)(12,20,22,24)(17,23,21,19);(2,5)(6,8)(9,11,14,15)(10,16,13,18)(12,19,22,23)(17,20,21,24);(9,12)(10,17)(11,19)(13,21)(14,22)(15,23)(16,20)(18,24);(1,3,4,7)(2,6,5,8)(9,13,14,10)(11,18,15,16)(12,21,22,17)(19,24,23,20);(9,11,14,15)(10,18,13,16)(12,19,22,23)(17,24,21,20)Polycyclic presentation:
2^8|a^2=h,b^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 | #6576730, #6576731, #6576742, … |
| maximal normal subgroup of | #6576730, #6576731, #6576742, … |
| maximal quotient of | #6371021, #6371817, #6371819, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53575);