Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419800 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23387 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1264 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)(5,6,9,15,11,16,10,27)(7,17,20,31,22,30,21,12)(8,14,24,34,25,33,26,13)(18,35,36,29,19,28,23,32);(5,7,9,21,11,22,10,20)(6,12,15,30,16,31,27,17)(8,23,24,19,25,36,26,18)(13,32,14,28,34,29,33,35);(5,8)(6,13)(7,18,22,19)(9,26)(10,24)(11,25)(12,28,31,35)(14,27)(15,33)(16,34)(17,29,30,32)(20,23,21,36);(2,4)(5,8)(6,14)(7,19,22,18)(9,24)(10,26)(11,25)(12,29,31,32)(13,27)(15,34)(16,33)(17,28,30,35)(20,23,21,36)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=efi,d^2=efi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339195);