Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#121 |
| prime radical | C22×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,10,16,11,15,9,14)(7,12,19,27,18,26,20,28)(8,21,22,13,23,29,24,30)(17,31,34,25,32,35,33,36);(5,7,10,20,11,18,9,19)(6,12,14,27,15,26,16,28)(8,17,24,34,23,32,22,33)(13,25,29,31,30,36,21,35);(2,4)(5,8)(6,13)(7,17)(9,22)(10,24)(11,23)(12,25)(14,29)(15,30)(16,21)(18,32)(19,33)(20,34)(26,36)(27,31)(28,35);(7,18)(12,26)(17,32)(19,20)(25,36)(27,28)(31,35)(33,34);(5,9,11,10)(6,14,15,16)(7,19,18,20)(8,22,23,24)(12,27,26,28)(13,29,30,21)(17,33,32,34)(25,31,36,35)Polycyclic presentation:
2^9|a^2=h,b^2=fh,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,d^b=di,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72237505, #72237516, #72237518, … |
| maximal normal subgroup of | #72237505, #72237516, #72237518, … |
| maximal quotient of | #72249680, #72257382, #72257446, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278732);