Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2104 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #401 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1443 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1434 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)(7,15,17,11)(8,12,20,27)(9,14)(10,13)(16,26)(18,31,33,25)(19,24)(21,28,22,30)(23,29)(32,36)(34,35);(5,7,9,18,10,17,23,33)(6,11,14,25,13,15,29,31)(8,19,21,34,20,16,22,32)(12,26,28,36,27,24,30,35);(2,4)(5,8)(6,12,13,27)(7,16)(9,22)(10,20)(11,24,15,26)(14,30,29,28)(17,19)(18,34)(21,23)(25,36,31,35)(32,33)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,f^b=fi,g^b=gi,e^c=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30981);