Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #2541 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10,14,12)(11,16,15,13)(18,20)(19,23);(2,5)(6,7)(9,11)(10,13)(12,16)(14,15)(17,18,21,20)(19,24,23,22);(9,12,14,10)(11,16,15,13)(17,19,21,23)(18,22,20,24);(1,2)(3,6)(4,5)(7,8)(17,18,21,20)(19,22,23,24);(1,3,4,8)(2,6,5,7)(9,13,14,16)(10,15,12,11)(17,20,21,18)(19,24,23,22)Polycyclic presentation:
2^8|a^2=fgh,b^2=g,c^2=fh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #7091684, #7091694, #7091700, … |
| maximal normal subgroup of | #7091684, #7091694, #7091700, … |
| maximal quotient of | #531799, #531937, #532065, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52671);