Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394127 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,14,11,9,27,15)(7,12,18,29,20,17,19,30)(8,21,24,23,26,32,25,13)(16,33,35,34,36,31,22,28);(1,2,3,4)(5,7)(6,12)(8,22)(9,17)(10,18)(11,20)(13,31)(14,29)(15,30)(16,24)(19,27)(21,28)(23,33)(25,36)(26,35)(32,34);(5,8,11,26)(6,13,9,23)(7,16,20,36)(10,25,27,24)(12,28,17,34)(14,32,15,21)(18,22,19,35)(29,31,30,33);(5,9)(6,11)(7,17)(8,23)(10,15)(12,20)(13,26)(14,27)(16,34)(18,30)(19,29)(21,25)(22,33)(24,32)(28,36)(31,35)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302810);