Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61569 |
| 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 | C2×#548 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1047 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,17,18,24)(12,20,22,15)(14,21,23,19);(1,3)(2,5)(4,8)(6,7)(11,18)(14,23)(17,24)(19,21);(9,11,13,18)(10,14,16,23)(12,21,22,19)(15,24,20,17);(3,8)(5,7);(2,6)(5,7)Polycyclic presentation:
2^8|a^2=h,c^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,c^b=ch,d^b=df,g^c=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | #245569, #245572, #245593, … |
| maximal normal subgroup of | #245569, #245572, #245593, … |
| maximal quotient of | #6386010, #6387596, #6387607, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54561);