Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #397 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1440 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,16,11,14,9,15)(7,17,19,32,20,12,26,28)(8,21,24,13,25,31,23,30)(18,33,34,27,22,36,35,29);(5,7)(6,12)(8,22,25,18)(9,19)(10,26)(11,20)(13,29,30,27)(14,17)(15,28)(16,32)(21,33,31,36)(23,35,24,34);(2,4)(5,8)(6,13)(7,18)(9,24)(10,23)(11,25)(12,27)(14,30)(15,31)(16,21)(17,29)(19,35)(20,22)(26,34)(28,33)(32,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,f^b=fi,g^b=gi,e^c=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237648, #72237662, #72237668, … |
| maximal normal subgroup of | #72237648, #72237662, #72237668, … |
| maximal quotient of | #72239269, #72239273, #72239275, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30814);