Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514160 |
| 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 | |
3089 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,11,16)(7,17,13,27)(8,12,24,32)(9,25,28,19)(10,15,29,14)(18,30,33,36)(20,21,35,26)(22,34,23,31);(1,2)(3,4)(5,7,11,13)(6,9,16,28)(8,20,24,35)(10,25,29,19)(12,30,32,36)(14,27,15,17)(18,22,33,23)(21,31,26,34);(2,4);(2,4)(5,8)(6,12)(7,18)(9,26)(10,22)(11,24)(13,33)(14,31)(15,34)(16,32)(17,30)(19,35)(20,25)(21,28)(23,29)(27,36);(5,9)(6,13)(7,16)(8,21)(10,27)(11,28)(12,18)(14,19)(15,25)(17,29)(20,31)(22,30)(23,36)(24,26)(32,33)(34,35)Polycyclic presentation:
2^9|a^2=i,b^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=cg,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, 7421358);