Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1859 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,11,15,12,9,10,16)(7,17,20,13,22,30,21,19)(8,23,26,14,28,32,27,25)(18,29,24,31,35,33,34,36);(1,3,4,2)(5,7)(6,13)(8,24)(9,19)(10,21)(11,20)(12,22)(14,31)(15,30)(16,17)(18,27)(23,29)(25,36)(26,35)(28,34)(32,33);(5,8)(6,14)(7,18)(9,25)(10,26)(11,27)(12,28)(13,29)(15,23)(16,32)(17,31)(19,33)(20,34)(21,24)(22,35)(30,36);(5,9,12,6)(7,19,22,13)(8,25,28,14)(10,15,11,16)(17,21,30,20)(18,33,35,29)(23,27,32,26)(24,36,34,31)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272371);