Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1436 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1730 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)(7,17)(8,21,24,30)(9,15)(10,14)(11,16)(12,19)(13,23,29,22)(18,26,34,36)(20,27)(25,28)(31,33,35,32);(2,4)(5,7,10,19,11,20,9,25)(6,12,14,27,16,28,15,17)(8,18,22,33,24,34,23,32)(13,26,30,35,29,36,21,31);(5,8)(6,13,16,29)(7,18)(9,23)(10,22)(11,24)(12,26,28,36)(14,30,15,21)(17,31,27,35)(19,33)(20,34)(25,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,h^a=hi,d^b=di,e^b=ei,d^c=di,f^c=fi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72250586, #72250603, #72250605, … |
| maximal normal subgroup of | #72250586, #72250603, #72250605, … |
| maximal quotient of | C2×#123501 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30697);