Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 262144 |
| 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)(13,14)(15,18)(16,20)(17,25)(19,22)(21,27)(23,28)(24,26);(5,6,8,10)(7,11,12,9)(13,15,19,16)(14,20,22,18)(17,23,27,24)(21,28,25,26);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,16,19,15)(14,20,22,18)(17,24,27,23)(21,28,25,26);(13,17,19,27)(14,21,22,25)(15,23,16,24)(18,26,20,28);(5,7,8,12)(6,9,10,11)(13,18,19,20)(14,15,22,16)(17,26,27,28)(21,23,25,24)Polycyclic presentation:
2^8|b^2=fh,c^2=fg,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | #98450, #130692, #130890, … |
| maximal normal subgroup of | #98450, #130692, #130890, … |
| maximal quotient of | #521621, #522254, #522908, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32161);