Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18346 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,10)(8,15,16,11)(9,12)(13,17,19,20)(14,18)(21,22,25,27,26,30,24,29)(23,31,33,28,34,35,32,36);(1,2,3,4)(5,7)(6,10)(8,13)(9,14)(11,17)(12,18)(15,20)(16,19)(22,27)(24,25)(28,31)(29,30)(32,33)(35,36);(5,8)(6,11)(7,13)(9,16)(10,17)(12,15)(14,19)(18,20)(21,23,26,34)(22,28,30,36)(24,33,25,32)(27,31,29,35);(6,12)(10,18)(11,15)(17,20)(21,24,26,25)(22,27,30,29)(23,32,34,33)(28,35,36,31)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=h,d^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=di,f^a=fhi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 102520);