Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
1174 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)(5,6,9,8)(7,11,12,10)(13,14)(15,19)(16,24)(17,18)(20,25)(21,22)(23,28)(26,27);(1,2,3,4)(6,8)(10,11)(14,18)(15,21)(16,25)(26,28);(5,7,9,12)(6,10,8,11)(13,14)(15,22)(16,20)(17,18)(19,21)(23,28)(24,25)(26,27);(1,2)(3,4)(5,8,9,6)(7,11,12,10)(13,15)(14,19)(16,23)(17,21)(18,22)(20,26)(24,28)(25,27);(13,16,17,25)(14,20,18,24)(15,23,21,27)(19,26,22,28)Polycyclic presentation:
2^8|a^2=g,b^2=fgh,c^2=g,d^2=g,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 | #7001042, #7001046, #7001049, … |
| maximal normal subgroup of | #7001042, #7001046, #7001049, … |
| maximal quotient of | #535320, #535487, #535949, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38304);