Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C22×#562 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 36 points:
(5,6)(7,14)(8,18)(9,12)(10,13)(11,17)(15,19)(16,20)(21,22)(23,28)(24,27)(25,29)(26,31)(30,35)(32,33)(34,36);(5,7,10,17)(6,11,13,14)(8,15,9,16)(12,19,18,20)(21,22)(23,28)(24,32)(25,29)(26,31)(27,33)(30,34)(35,36);(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(21,23)(22,26)(24,30)(25,31)(27,34)(28,29)(32,35)(33,36);(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(21,24)(22,27)(23,30)(25,33)(26,34)(28,35)(29,32)(31,36);(1,2,3,4)(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(21,23)(22,28)(24,30)(25,31)(26,29)(27,35)(32,34)(33,36);(5,6)(7,11)(8,12)(9,18)(10,13)(14,17)(15,19)(16,20)(21,22)(23,26)(24,27)(25,29)(28,31)(30,34)(32,33)(35,36)Polycyclic presentation:
2^9|b^2=g,c^2=g,e^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7534607);