Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394055 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3020 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2109 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,16)(7,18,21,13)(8,23)(9,17,11,15)(12,22,28,20)(14,25)(19,27)(24,31)(26,30)(29,33)(32,34)(35,36);(1,2,3,4)(5,7,9,20,10,21,11,22)(6,12,15,18,16,28,17,13)(8,19,24,33,26,35,25,34)(14,27,30,32,31,36,23,29);(5,7,10,21)(6,13,16,18)(8,19,26,35)(9,22,11,20)(12,15,28,17)(14,29,31,32)(23,27,30,36)(24,34,25,33);(5,8)(6,14)(7,19)(9,24)(10,26)(11,25)(12,27)(13,29)(15,30)(16,31)(17,23)(18,32)(20,33)(21,35)(22,34)(28,36)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^c=ei,g^c=gi,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302829);