Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | Aut(G) order 524288 |
| 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×#532 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(6,8)(7,11)(13,14)(15,20)(16,24)(17,22)(18,19)(21,26)(23,25)(27,28);(5,6,9,8)(7,10,11,12)(14,19)(15,22)(21,24)(23,28);(1,2)(3,4)(7,11)(10,12)(13,15,18,22)(14,20,19,17)(16,23,26,28)(21,27,24,25);(5,7)(6,10)(8,12)(9,11)(13,16,18,26)(14,21,19,24)(15,23,22,28)(17,25,20,27);(5,8,9,6)(7,12,11,10)(13,17,18,20)(14,22,19,15)(16,25,26,27)(21,28,24,23)Polycyclic presentation:
2^8|a^2=g,b^2=fh,c^2=h,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #112000, #112036, #112062, … |
| maximal normal subgroup of | #112000, #112036, #112062, … |
| maximal quotient of | #556340, #556377, #556392, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30622);