Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62265 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1010 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)(10,16)(11,18)(13,15)(17,24)(19,23)(20,21);(1,3)(2,5)(4,8)(6,7)(9,10)(11,17)(12,15)(13,22)(14,16)(18,19)(20,24)(21,23);(9,11,13,20,14,21,15,18)(10,17,22,24,16,23,12,19);(3,8)(5,7)(9,12)(10,13)(11,19)(14,22)(15,16)(17,20)(18,23)(21,24);(2,6)(5,7)(9,13,14,15)(10,12,16,22)(11,20,21,18)(17,19,23,24)Polycyclic presentation:
2^8|c^2=gh,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #130234, #130237, #130435, … |
| maximal normal subgroup of | #130234, #130237, #130435, … |
| maximal quotient of | #6386764, #6389171, #6389318, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55316);