Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1027 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)(5,6,8,10)(7,9,12,11)(14,18)(15,21)(20,23)(22,27);(5,6,8,10)(7,11,12,9)(13,14)(15,19)(16,23)(17,18)(20,24)(21,25)(22,28)(26,27);(1,2,3,4)(5,6)(7,9)(8,10)(11,12);(13,15,17,21)(14,19,18,25)(16,22,24,27)(20,26,23,28);(5,7,8,12)(6,9,10,11)(13,16,17,24)(14,20,18,23)(15,22,21,27)(19,26,25,28)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=gh,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | #98408, #130560, #130569, … |
| maximal normal subgroup of | #98408, #130560, #130569, … |
| maximal quotient of | #521494, #522146, #522652, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32732);