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 | C2×#3529 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6746 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)(7,12)(8,11)(9,13)(10,15)(14,19)(16,17)(18,20)(21,22)(23,30)(24,34)(25,28)(26,29)(27,33)(31,35)(32,36);(5,6)(7,12)(8,16)(9,13)(10,15)(11,17)(14,18)(19,20)(21,23,26,33)(22,27,29,30)(24,31,25,32)(28,35,34,36);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,24,26,25)(22,28,29,34)(23,31,33,32)(27,35,30,36);(5,8)(6,11)(7,14)(9,17)(10,18)(12,19)(13,16)(15,20)(21,23)(22,27)(24,31)(25,32)(26,33)(28,35)(29,30)(34,36);(1,2,3,4)(5,7)(6,12)(8,14)(9,15)(10,13)(11,19)(16,18)(17,20)(21,22)(23,27)(24,28)(25,34)(26,29)(30,33)(31,35)(32,36);(5,6)(7,10)(8,11)(9,13)(12,15)(14,18)(16,17)(19,20)(21,25,26,24)(22,28,29,34)(23,32,33,31)(27,35,30,36)Polycyclic presentation:
2^9|b^2=g,c^2=g,e^2=i,f^2=g,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, 7534608);