Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437602 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60366 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3063 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(5,6)(7,12)(8,14)(9,13)(10,16)(11,15)(17,27)(18,25)(19,28)(20,26)(21,30)(22,29)(23,31)(24,32)(33,36)(34,35);(1,2,4,3)(5,7,11,20)(6,12,15,26)(8,21,23,33)(9,19,10,17)(13,28,16,27)(14,30,31,36)(18,24,34,22)(25,32,35,29);(1,3,4,2)(6,13)(7,17)(9,10)(14,29)(15,16)(18,21)(19,20)(22,24)(27,28)(30,36)(31,32)(33,34);(5,8)(6,14)(7,18)(9,22)(10,24)(11,23)(12,25)(13,29)(15,31)(16,32)(17,21)(19,33)(20,34)(26,35)(27,30)(28,36);(6,15)(12,26)(13,16)(14,31)(25,35)(27,28)(29,32)(30,36)Polycyclic presentation:
2^9|b^2=fi,c^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346050);