Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1089 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)(17,18)(19,21)(20,23)(22,24);(1,2,4,6)(3,5,8,7)(10,13)(11,15)(17,19,20,24)(18,21,23,22);(9,10,12,13)(11,14,15,16)(17,19)(18,22)(20,24)(21,23);(1,3)(2,5)(4,8)(6,7);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,23)(21,24)Polycyclic presentation:
2^8|b^2=fg,c^2=gh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#38109, #7635426, #7635455, … |
| maximal normal subgroup of | C2×#38109, #7635426, #7635455, … |
| maximal quotient of | #521704, #522321, #522826, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 31097);