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 | C2×#562 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1018 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)(5,6,9,10)(7,8,12,11)(13,14)(15,16)(17,25)(18,19)(20,23)(21,26)(22,27)(24,28);(6,10)(8,11)(14,19)(16,20)(21,25)(24,27);(1,2,3,4)(5,6,9,10)(7,11,12,8)(13,15)(14,20)(16,19)(17,22)(18,23)(21,27)(24,25)(26,28);(5,7)(6,11)(8,10)(9,12)(13,16,18,20)(14,15,19,23)(17,24,26,27)(21,22,25,28);(5,8,9,11)(6,7,10,12)(13,17,18,26)(14,21,19,25)(15,22,23,28)(16,24,20,27)Polycyclic presentation:
2^8|a^2=h,c^2=fg,d^2=fh,e^2=f,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 | #98478, #262182, #262184, … |
| maximal normal subgroup of | #98478, #262182, #262184, … |
| maximal quotient of | #557081, #557083, #557385, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39137);