Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | #392 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(6,9)(7,11)(13,14,18,22,17,21,16,20)(15,19,23,26,24,27,25,28);(5,6,8,9)(7,10,11,12)(13,14,16,20,17,21,18,22)(15,19,25,28,24,27,23,26);(2,4)(6,9)(10,12)(13,15)(14,19)(16,25)(17,24)(18,23)(20,28)(21,27)(22,26);(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,20,21,22)(15,23,24,25)(19,26,27,28)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=ceh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #209584, #209586, #209588, … |
| maximal normal subgroup of | #209584, #209586, #209588, … |
| maximal quotient of | #218530, #218531, #218532, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15128);