Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6443052 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60454 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,9,13,10,15,22,14)(7,16,19,11,18,25,32,24)(8,12,21,27,20,26,33,28)(17,29,30,23,31,34,36,35);(1,2)(3,4)(5,6)(7,11)(8,12)(9,14)(10,15)(13,22)(16,19)(17,23)(18,24)(20,26)(21,28)(25,32)(27,33)(29,30)(31,35)(34,36);(2,4)(5,7,9,19,10,18,22,32)(6,11,14,16,15,24,13,25)(8,17,21,30,20,31,33,36)(12,23,28,29,26,35,27,34);(2,4)(5,8)(6,12)(7,17)(9,21)(10,20)(11,23)(13,27)(14,28)(15,26)(16,29)(18,31)(19,30)(22,33)(24,35)(25,34)(32,36);(7,18)(8,20)(11,24)(12,26)(16,25)(19,32)(21,33)(27,28)Polycyclic presentation:
2^9|a^2=f,c^2=g,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350248);