Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 262144 |
| 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×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
977 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(7,11)(9,12)(13,14,18,17)(15,19,23,22)(16,24,25,20)(21,27,28,26);(2,3)(5,6,8,10)(7,9,11,12)(14,17)(19,22)(20,24)(26,27);(1,2,4,3)(5,7,8,11)(6,9,10,12)(13,15,18,23)(14,19,17,22)(16,21,25,28)(20,26,24,27);(13,16,18,25)(14,20,17,24)(15,21,23,28)(19,26,22,27);(1,3,4,2)(13,17,18,14)(15,22,23,19)(16,24,25,20)(21,27,28,26)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=fgh,d^2=f,e^2=fh,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 | #98432, #258612, #258614, … |
| maximal normal subgroup of | #98432, #258612, #258614, … |
| maximal quotient of | #521436, #522152, #522706, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32367);