Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1437 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2482 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)(7,16,19,12)(8,20)(9,23)(10,15)(11,14)(13,21)(17,32,34,36)(18,27,24,28)(22,31)(25,30)(26,33,29,35);(2,4)(5,7,10,18,11,19,9,24)(6,12,15,28,14,16,23,27)(8,17,21,35,22,34,25,33)(13,29,31,32,30,26,20,36);(5,8)(6,13)(7,17)(9,25)(10,21)(11,22)(12,26)(14,30)(15,31)(16,29)(18,35)(19,34)(20,23)(24,33)(27,32)(28,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,e^b=ei,g^b=gi,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237174, #72237175, #72237188, … |
| maximal normal subgroup of | #72237174, #72237175, #72237188, … |
| maximal quotient of | #72239031, #72249707, #72257440, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30759);