Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9467 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C26 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4)(5,6)(7,9)(8,11)(10,12)(17,20);(6,9,11,10)(13,14)(15,17)(16,18)(19,20);(13,15)(14,17)(16,19)(18,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,e^b=ei,d^c=di98 conjugacy classes — view the full character table.
| maximal subgroup of | #72241252, #72283750, #72283877, … |
| maximal normal subgroup of | #72241252, #72283750, #72283877, … |
| maximal quotient of | #72241266, #72267432, #72285356, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 15378);