Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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 | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1211 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(7,11)(10,12)(13,14,19,22)(15,17,16,20)(18,26,27,21)(23,28,24,25);(6,9)(10,12)(13,15,19,16)(14,20,22,17)(18,23,27,24)(21,28,26,25);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,19,15)(14,20,22,17)(18,24,27,23)(21,28,26,25);(5,6)(7,10)(8,9)(11,12)(13,17,19,20)(14,15,22,16)(18,25,27,28)(21,23,26,24);(5,7)(6,10)(8,11)(9,12)(13,18)(14,21)(15,23)(16,24)(17,25)(19,27)(20,28)(22,26)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=f,d^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97221, #131972, #132158, … |
| maximal normal subgroup of | #97221, #131972, #132158, … |
| maximal quotient of | #553731, #553994, #554191, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29699);