Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| 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 | #2539 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1123 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,10,9)(7,12,8,11)(13,14,18,22)(15,19,23,28)(16,20,26,25)(17,21,27,24);(5,7,10,8)(6,11,9,12)(15,23)(17,27)(19,28)(21,24);(1,2,3,4)(5,8,10,7)(6,12,9,11)(13,15)(14,19)(16,21)(17,25)(18,23)(20,27)(22,28)(24,26);(13,16)(14,20)(15,24)(17,19)(18,26)(21,23)(22,25)(27,28);(5,9,10,6)(7,12,8,11)(13,17)(14,21)(15,25)(16,19)(18,27)(20,23)(22,24)(26,28)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=g,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #132165, #133738, #213767, … |
| maximal normal subgroup of | #132165, #133738, #213767, … |
| maximal quotient of | #515353, #515424, #515572, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28457);