Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2361 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,12,16,11,17,10,18)(7,19,24,32,23,35,22,13)(8,25,21,36,28,14,27,31)(9,29,30,33,26,34,20,15);(5,7,10,22,11,23,12,24)(6,13,16,19,17,32,18,35)(8,26,27,30,28,9,21,20)(14,15,31,29,25,33,36,34);(2,4)(5,8,12,21,11,28,10,27)(6,14,16,31,17,25,18,36)(7,20,24,9,23,30,22,26)(13,29,19,33,32,34,35,15);(5,9,10,20,11,26,12,30)(6,15,16,29,17,33,18,34)(7,21,22,8,23,27,24,28)(13,31,19,25,32,36,35,14)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=h,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281072);