Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63011 |
| 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×#3482 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,3,4)(6,12)(7,8)(14,20)(15,16)(23,30)(27,31)(32,35)(33,36);(5,6)(7,9)(8,13)(10,14)(11,12)(15,17)(16,18)(19,20)(21,22,26,29)(23,31,30,27)(24,25,28,34)(32,36,35,33);(5,7,11,8)(6,9,12,13)(10,15,19,16)(14,17,20,18)(21,23,26,30)(22,27,29,31)(24,32,28,35)(25,33,34,36);(5,8,11,7)(6,13,12,9)(10,16,19,15)(14,18,20,17);(5,9,11,13)(6,7,12,8)(10,18,19,17)(14,16,20,15)(21,24)(22,25)(23,32)(26,28)(27,33)(29,34)(30,35)(31,36);(5,10)(6,14)(7,15)(8,16)(9,17)(11,19)(12,20)(13,18)(21,25)(22,28)(23,33)(24,29)(26,34)(27,35)(30,36)(31,32)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,f^e=fh152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7534426);