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 | #2404 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1201 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)(5,6,9,12)(7,8,11,10)(13,14,18,20)(15,21,22,17)(16,19,26,28)(23,27,24,25);(5,7,9,11)(6,10,12,8)(15,22)(17,21)(23,24)(25,27);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,17)(16,24)(18,22)(19,27)(20,21)(23,26)(25,28);(13,16)(14,19)(15,23)(17,25)(18,26)(20,28)(21,27)(22,24);(5,8,9,10)(6,11,12,7)(13,17)(14,15)(16,25)(18,21)(19,23)(20,22)(24,28)(26,27)Polycyclic presentation:
2^8|a^2=fh,b^2=f,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 | #132089, #132149, #213735, … |
| maximal normal subgroup of | #132089, #132149, #213735, … |
| maximal quotient of | #515317, #515424, #515545, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28442);