Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1001 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)(13,14)(15,21)(16,24)(17,18)(19,23)(20,25)(22,26)(27,28);(5,6,9,12)(7,11,8,10)(14,18)(19,21)(20,24)(26,27);(1,2,3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,27)(25,28);(13,16,17,25)(14,20,18,24)(15,22,23,28)(19,26,21,27);(5,8,9,7)(6,11,12,10)(13,14)(15,19)(16,20)(17,18)(21,23)(22,26)(24,25)(27,28)Polycyclic presentation:
2^8|b^2=h,c^2=fgh,d^2=f,e^2=h,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 | #497597, #497606, #6535048, … |
| maximal normal subgroup of | #497597, #497606, #6535048, … |
| maximal quotient of | #521495, #522198, #522752, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32601);