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) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#534 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1251 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)(5,6)(7,11)(8,9)(10,12)(16,23)(20,25)(22,27)(26,28);(6,9)(10,11)(14,18)(15,21)(20,25)(22,27);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28);(5,7,8,12)(6,10,9,11)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28);(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)Polycyclic presentation:
2^8|a^2=g,c^2=fh,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 | #97241, #132103, #133091, … |
| maximal normal subgroup of | #97241, #132103, #133091, … |
| maximal quotient of | #553727, #553989, #554185, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29957);