Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9443 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
2630 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)(7,11)(8,10)(9,12)(14,17)(15,19)(18,20);(5,7)(6,9,10,11)(8,12)(13,14)(15,20)(16,18)(17,19);(15,19)(17,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,d^b=di,e^b=eh,d^c=dh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72226368, #72258003 |
| maximal normal subgroup of | #72226368, #72258003 |
| maximal quotient of | #72225897 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13710);