Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24987 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1256 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)(5,6,10,16,11,13,9,15)(7,12,20,24,19,26,17,29)(8,14,22,28,21,27,23,30)(18,32,31,35,34,25,33,36);(2,4)(5,7,9,20,11,19,10,17)(6,12,16,29,13,26,15,24)(8,18,22,33,21,34,23,31)(14,25,30,36,27,32,28,35);(6,13)(7,17)(9,10)(12,24)(14,27)(18,31)(19,20)(22,23)(25,35)(26,29)(32,36)(33,34);(2,4)(5,8,11,21)(6,14)(7,18,19,34)(9,22,10,23)(12,25)(13,27)(15,28)(16,30)(17,31,20,33)(24,35)(26,32)(29,36)Polycyclic presentation:
2^9|a^2=h,b^2=fg,d^2=fi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,h^b=hi,g^c=gi,h^c=hi,e^d=ei,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269703);