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 | C22⋊#95 |
| 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:
(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);(1,2,3,4)(5,7)(6,12)(8,18)(9,25)(10,19)(11,20)(13,26)(14,27)(15,17)(16,28)(21,31)(22,33)(23,32)(24,34)(29,36)(30,35);(5,8,10,22,11,24,9,23)(6,13,15,21,16,29,14,30)(7,18,19,33,20,34,25,32)(12,26,17,31,28,36,27,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=i,e^2=i,h^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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30696);