Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 131072 |
| 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 | |
1116 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)(13,14,17,20)(15,21,23,18)(16,19,25,24)(22,27,28,26);(2,4)(5,6,8,10)(7,12,11,9)(13,14)(15,18)(16,24)(17,20)(19,25)(21,23)(22,27)(26,28);(1,2)(3,4)(5,7,8,11)(6,9,10,12);(13,15,17,23)(14,18,20,21)(16,22,25,28)(19,26,24,27);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16)(14,19)(15,22)(17,25)(18,26)(20,24)(21,27)(23,28)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=g,d^2=f,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 | #6536111, #6536113, #6536115, … |
| maximal normal subgroup of | #6536111, #6536113, #6536115, … |
| maximal quotient of | #521363, #522097, #522607, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30683);