Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C25 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1067 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(6,9)(7,11)(13,14)(15,18)(16,21)(17,22)(19,23)(20,25)(24,26)(27,28);(6,9)(10,12)(14,15)(16,23)(20,22)(24,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,14)(16,21,23,19)(17,22,25,20)(24,27,28,26);(5,7,8,11)(6,10,9,12)(13,16)(14,19)(15,21)(17,24)(18,23)(20,26)(22,27)(25,28);(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)Polycyclic presentation:
2^8|a^2=h,c^2=f,d^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df76 conjugacy classes — view the full character table.
| maximal subgroup of | #107184, #107258, #190197, … |
| maximal normal subgroup of | #107184, #107258, #190197, … |
| maximal quotient of | #513790, #513815, #513818, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27369);