Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11247 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #387 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1441 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1586 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)(7,17,20,12)(8,22)(9,16)(10,15)(11,14)(13,24)(18,34,36,30)(19,28,21,29)(23,27,35,33)(25,31)(26,32);(1,2)(3,4)(5,7,10,21,11,20,9,19)(6,12,15,28,14,17,16,29)(8,23,24,18,26,35,25,36)(13,30,32,27,31,34,22,33);(2,4)(5,8,9,24,11,26,10,25)(6,13,15,22,14,31,16,32)(7,18,19,35,20,36,21,23)(12,27,28,30,17,33,29,34)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,e^2=i,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,g^b=gi,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72252097, #72252100, #72252110, … |
| maximal normal subgroup of | #72252097, #72252100, #72252110, … |
| maximal quotient of | C2×#123976 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31172);