Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
6510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,3,4)(9,10)(11,12)(13,14)(15,20)(16,18)(17,19);(5,6,7,8)(13,15,17,16)(14,18,19,20);(2,4)(6,8)(13,14)(15,20)(16,18)(17,19);(10,12)(14,19)(18,20);(13,16,17,15)(14,18,19,20)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^c=df104 conjugacy classes — view the full character table.
| maximal subgroup of | #72264085, #72264116, #72264959, … |
| maximal normal subgroup of | #72264085, #72264116, #72264959, … |
| maximal quotient of | #72264400, #72264688, C2×#611060 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 518256);