Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418464 |
| 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 | #60644 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1891 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)(7,11)(8,19)(9,25)(10,15)(12,16)(13,23)(14,26)(17,24)(18,28)(20,22)(21,33)(27,30)(29,35)(31,36)(32,34);(1,2,3,4)(5,7,11,6,12,15,10,16)(8,20,23,17,24,13,22,19)(9,14,27,18,28,30,26,25)(21,31,36,33,35,32,34,29);(5,8,11,23,12,24,10,22)(6,13,15,19,16,20,7,17)(9,21,26,34,28,35,27,36)(14,29,25,32,30,33,18,31);(2,4)(5,9)(6,14)(7,18)(8,21)(10,27)(11,26)(12,28)(13,29)(15,25)(16,30)(17,31)(19,32)(20,33)(22,36)(23,34)(24,35)Polycyclic presentation:
2^9|b^2=ei,c^2=f,e^2=i,f^2=i,g^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,g^b=gi,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, 330121);