Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514172 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3045 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,10,12)(7,16,20,13)(8,21,24,32)(9,11,26,15)(14,25,28,23)(17,27,19,30)(18,34,22,36)(29,33,31,35);(1,2)(3,4)(6,11)(7,17)(9,26)(12,15)(14,21)(18,33)(19,20)(22,35)(23,25)(27,30)(28,32)(34,36);(2,4)(6,12)(11,15)(13,16)(14,28)(21,32)(27,30)(29,31)(34,36);(2,4)(5,7)(6,13)(8,22)(9,19)(10,20)(11,27)(12,16)(14,31)(15,30)(17,26)(18,24)(21,34)(23,35)(25,33)(28,29)(32,36);(5,8)(6,14)(7,18)(9,25)(10,24)(11,21)(12,28)(13,29)(15,32)(16,31)(17,33)(19,35)(20,22)(23,26)(27,36)(30,34)Polycyclic presentation:
2^9|a^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cgi,d^b=dh,f^b=fi,h^b=hi,e^d=ei,h^d=hi,f^e=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421381);