Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1438 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,15)(8,19,21,12)(9,22)(10,14)(11,17)(13,23)(16,28,31,36)(18,29)(20,27,34,33)(24,35,26,30)(25,32);(1,2)(3,4)(5,7)(6,11,14,25)(8,16)(9,17)(10,18)(12,26,19,24)(13,15,22,29)(20,30)(21,31)(23,32)(27,36,33,28)(34,35);(2,4)(5,8)(6,12)(7,16)(9,20)(10,21)(11,24)(13,27)(14,19)(15,28)(17,30)(18,31)(22,33)(23,34)(25,26)(29,36)(32,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,h^a=hi,c^b=cf,e^b=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239305, #72239777, #72239913, … |
| maximal normal subgroup of | #72239305, #72239777, #72239913, … |
| maximal quotient of | #72226732, #72239268, #72239275, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30732);