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×#540 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,4,3)(5,6)(7,14)(8,11)(9,12)(10,13)(15,20)(16,18)(17,19)(21,22,26,27)(23,28,30,25)(24,32,34,29)(31,33,36,35);(5,6)(7,14)(8,16)(9,12)(10,13)(11,18)(15,17)(19,20)(22,27)(23,30)(29,32)(31,36);(5,7,10,12)(6,9,13,14)(8,15,18,19)(11,17,16,20)(21,23)(22,28)(24,31)(25,27)(26,30)(29,35)(32,33)(34,36);(5,8)(6,11)(7,15)(9,17)(10,18)(12,19)(13,16)(14,20)(21,24,26,34)(22,29,27,32)(23,31,30,36)(25,33,28,35);(1,3,4,2)(5,6)(7,9)(8,11)(10,13)(12,14)(15,17)(16,18)(19,20);(5,9)(6,12)(7,13)(8,17)(10,14)(11,19)(15,16)(18,20)(21,25,26,28)(22,23,27,30)(24,33,34,35)(29,31,32,36)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=g,e^2=gi,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, 7534709);