Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1188 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(21,22,27,32,26,31,25,30)(23,28,33,36,24,35,34,29);(5,7,8,12)(6,9,10,11)(13,14)(15,18)(16,17)(19,20)(21,23)(22,28)(24,26)(25,34)(27,33)(29,30)(31,35)(32,36);(2,4)(14,17)(18,20)(21,24,26,23)(22,29,31,36)(25,34,27,33)(28,32,35,30);(6,10)(9,11)(13,15,16,19)(14,18,17,20)(21,25,26,27)(22,30,31,32)(23,33,24,34)(28,36,35,29)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=i,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 270509);