Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1115 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,9)(10,12)(13,14,19,18)(15,22,17,20)(16,21,26,25)(23,28,24,27);(2,3)(6,9)(7,11)(13,15,19,17)(14,20,18,22)(16,24,26,23)(21,28,25,27);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,16,19,26)(14,21,18,25)(15,23,17,24)(20,27,22,28);(13,17,19,15)(14,22,18,20)(16,24,26,23)(21,28,25,27);(1,3,4,2)(5,7)(6,10)(8,11)(9,12)(13,18,19,14)(15,22,17,20)(16,25,26,21)(23,28,24,27)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=gh,d^2=f,e^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 | #6533957, #6533961, #6533963, … |
| maximal normal subgroup of | #6533957, #6533961, #6533963, … |
| maximal quotient of | #521417, #522134, #522681, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32423);