Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418492 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×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 | |
1870 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)(7,19)(8,23)(9,24)(10,17)(11,16)(12,18)(13,21)(14,26)(15,29)(20,30)(22,36)(25,34)(27,28)(31,35)(32,33);(1,2,3,4)(5,7,10,21,12,22,11,32)(6,13,16,19,18,33,17,36)(8,24,25,35,27,20,26,15)(9,28,29,34,30,23,31,14);(5,8,11,26,12,27,10,25)(6,14,17,23,18,34,16,28)(7,20,21,15,22,24,32,35)(9,19,29,33,30,36,31,13);(2,4)(5,9,12,30)(6,15,18,35)(7,14,22,34)(8,19,27,36)(10,31,11,29)(13,26,33,25)(16,20,17,24)(21,23,32,28)Polycyclic presentation:
2^9|b^2=e,c^2=f,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^c=gi,e^d=ei,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330102);