Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418504 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60645 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1517 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)(5,6,11,17)(7,18,22,33)(8,23,26,31)(9,16,10,15)(12,27,30,21)(13,24,14,25)(19,28,36,32)(20,29,35,34);(5,7,9,21,11,22,10,27)(6,12,15,18,17,30,16,33)(8,19,24,35,26,36,25,20)(13,28,31,29,14,32,23,34);(5,8)(6,13)(7,19,22,36)(9,25)(10,24)(11,26)(12,28,30,32)(14,17)(15,23)(16,31)(18,34,33,29)(20,27,35,21);(2,4)(5,8)(6,14)(7,20)(9,25)(10,24)(11,26)(12,29)(13,17)(15,31)(16,23)(18,28)(19,27)(21,36)(22,35)(30,34)(32,33)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330478);