Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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 | C2×#540 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1079 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)(7,11)(10,12)(13,14)(15,21)(16,18)(17,24)(19,23)(20,25)(22,26)(27,28);(6,9)(10,12)(14,16)(19,21)(20,24)(26,27);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,21)(17,22)(18,23)(20,26)(24,27)(25,28);(5,6)(7,10)(8,9)(11,12)(13,16,18,14)(15,21,23,19)(17,24,25,20)(22,27,28,26);(5,7,8,11)(6,10,9,12)(13,17,18,25)(14,20,16,24)(15,22,23,28)(19,26,21,27)Polycyclic presentation:
2^8|c^2=fg,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #214046, #214652, #6481602, … |
| maximal normal subgroup of | #214046, #214652, #6481602, … |
| maximal quotient of | #553766, #553768, #554021, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30292);