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 | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1450 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)(5,6,10,14,11,7,9,13)(8,16,17,12,19,20,18,15)(21,22)(23,26)(24,31,32,27)(25,28)(29,33,35,36)(30,34);(1,2,3,4)(5,7,11,6)(8,15,19,12)(9,13,10,14)(16,17,20,18)(21,23)(22,26)(24,29)(25,30)(27,33)(28,34)(31,36)(32,35);(5,8,11,19)(6,12,7,15)(9,18,10,17)(13,16,14,20)(21,24)(22,27)(23,29)(25,32)(26,33)(28,31)(30,35)(34,36);(5,9,11,10)(6,13,7,14)(8,17,19,18)(12,20,15,16)(22,28)(26,34)(27,31)(33,36)Polycyclic presentation:
2^9|a^2=g,b^2=egh,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, 102521);