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) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1140 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)(13,14,19,24)(15,21,16,20)(17,18,23,22)(25,26,27,28);(5,6)(7,11)(8,9)(10,12)(13,15,19,16)(14,20,24,21)(17,25,23,27)(18,28,22,26);(1,2,3,4)(6,9)(10,11)(13,16,19,15)(14,21,24,20)(17,25,23,27)(18,28,22,26);(5,6)(7,10)(8,9)(11,12)(13,17,19,23)(14,22,24,18)(15,25,16,27)(20,26,21,28);(5,7,8,12)(6,10,9,11)(13,18,19,22)(14,23,24,17)(15,26,16,28)(20,25,21,27)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=f,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 | #132371, #248431, #248432, … |
| maximal normal subgroup of | #132371, #248431, #248432, … |
| maximal quotient of | #544521, #545080, #545509, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35469);