Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1177 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,10)(7,11,12,9)(13,14,18,17)(15,21,23,19)(16,24,26,20)(22,27,25,28);(5,7,8,12)(6,9,10,11)(14,17)(19,21)(20,24)(27,28);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,18,23)(14,19,17,21)(16,25,26,22)(20,28,24,27);(5,6)(7,9)(8,10)(11,12)(13,16,18,26)(14,20,17,24)(15,22,23,25)(19,27,21,28);(13,17,18,14)(15,21,23,19)(16,24,26,20)(22,28,25,27)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=fh,d^2=fh,e^2=fh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #132377, #248800, #248801, … |
| maximal normal subgroup of | #132377, #248800, #248801, … |
| maximal quotient of | #544416, #545006, #545376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35983);