Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1003 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)(5,6,10,9)(7,12,8,11)(13,14,18,22)(15,23,24,19)(16,20,27,25)(17,26,28,21);(5,7,10,8)(6,11,9,12)(15,24)(17,28)(19,23)(21,26);(1,2,3,4)(5,8,10,7)(6,11,9,12)(13,15,18,24)(14,19,22,23)(16,26,27,21)(17,25,28,20);(5,9,10,6)(7,11,8,12)(13,16)(14,20)(15,21)(17,23)(18,27)(19,28)(22,25)(24,26);(13,17,18,28)(14,21,22,26)(15,25,24,20)(16,23,27,19)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=fg,d^2=f,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #130704, #130865, #245733, … |
| maximal normal subgroup of | #130704, #130865, #245733, … |
| maximal quotient of | #535402, #535556, #536061, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38131);