Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #2920 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1584 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,8,5,7,11,14)(3,6,10,13,9,12,15,16)(17,18,22,25,23,19,21,27)(20,24,30,31,28,26,29,32);(17,19,23,18)(20,26,28,24)(21,25,22,27)(29,31,30,32);(1,3,4,10,5,9,11,15)(2,6,8,13,7,12,14,16)(17,20,22,30,23,28,21,29)(18,24,25,31,19,26,27,32);(3,9)(6,12)(10,15)(13,16);(2,7)(6,12)(8,14)(13,16)(20,28)(24,26)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,f^2=h,g^2=i,b^a=bf,e^a=ehi,f^a=fh,f^b=fh,d^c=dhi,e^c=eh98 conjugacy classes — view the full character table.
| maximal subgroup of | #72283124, #72306825, #72306829, … |
| maximal normal subgroup of | #72283124, #72306825, #72306829, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252597);