Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 262144 |
| 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×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1147 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)(6,9)(10,12)(14,17)(15,21)(20,24)(23,27);(1,2)(3,4)(13,14,18,17)(15,22,21,19)(16,24,25,20)(23,26,27,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,21)(14,19,17,22)(16,23,25,27)(20,26,24,28);(13,16,18,25)(14,20,17,24)(15,23,21,27)(19,26,22,28);(5,7,8,11)(6,10,9,12)(13,17,18,14)(15,22,21,19)(16,24,25,20)(23,28,27,26)Polycyclic presentation:
2^8|b^2=h,c^2=h,d^2=h,e^2=g,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #111680, #111962, #112921, … |
| maximal normal subgroup of | #111680, #111962, #112921, … |
| maximal quotient of | #516485, #516572, #516648, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28233);