Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 |
| prime radical | #2682 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1004 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)(6,7)(10,11)(14,15)(17,25)(19,21)(23,28);(6,7)(8,12)(13,14,18,15)(16,21,24,19)(17,20,25,22)(23,27,28,26);(1,2,3,4)(5,6,9,7)(8,10,12,11)(13,15,18,14)(16,21,24,19)(17,22,25,20)(23,27,28,26);(5,7,9,6)(8,11,12,10)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27);(5,8,9,12)(6,10,7,11)(13,17)(14,20)(15,22)(16,23)(18,25)(19,26)(21,27)(24,28)Polycyclic presentation:
2^8|b^2=f,c^2=g,d^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97210, #172181, #172182, … |
| maximal normal subgroup of | #97210, #172181, #172182, … |
| maximal quotient of | #557085, #557390, #557747, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39269);