Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13247 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1972 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
744 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)(5,6,10,15,12,8,24,16)(7,17,21,34,23,35,20,33)(9,27,28,14,11,30,29,26)(13,31,32,22,25,36,18,19);(2,4)(5,7,11,22,24,20,28,31,12,23,9,19,10,21,29,36)(6,13,14,17,15,32,30,34,8,25,26,35,16,18,27,33);(5,8)(6,12)(7,18,23,32)(9,26)(10,15)(11,14)(13,21,25,20)(16,24)(17,31,35,36)(19,33,22,34)(27,29)(28,30)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=efi,d^2=hi,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,h^a=hi,c^b=ch,e^b=ei,e^c=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250147, #72250154, #72250158, … |
| maximal normal subgroup of | #72250147, #72250154, #72250158, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54632);