Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20158 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,9,11,10,14,8,13)(7,12,17,18,16,19,15,20)(21,22,25,28)(23,29,30,36)(24,32,33,27)(26,34,35,31);(1,3,4,2)(6,11)(7,15)(8,9)(13,14)(16,17)(18,20)(23,30)(26,35)(29,36)(31,34);(7,16)(12,19)(15,17)(18,20)(21,23,24,31,25,30,33,34)(22,26,27,29,28,35,32,36);(5,7,8,15,10,16,9,17)(6,12,13,20,14,19,11,18)Polycyclic presentation:
2^9|a^2=g,b^2=egi,c^2=f,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,f^a=fi,c^b=ci,d^b=de,e^b=eh,g^b=gh,d^c=dh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 138847);