Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519063 |
| 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 | |
2501 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)(5,6,11,16)(7,13,19,27)(8,20,23,34)(9,24,10,15)(12,25,28,18)(14,22,30,21)(17,31,33,26)(29,35,36,32);(1,2)(3,4)(5,7,10,25,11,19,9,18)(6,12,24,27,16,28,15,13)(8,17,22,35,23,33,21,32)(14,26,34,36,30,31,20,29);(2,4);(2,4)(5,7)(6,13)(8,17)(9,18)(10,25)(11,19)(12,24)(14,29)(15,28)(16,27)(20,31)(21,32)(22,35)(23,33)(26,34)(30,36);(5,8,10,22,11,23,9,21)(6,14,24,34,16,30,15,20)(7,17,25,35,19,33,18,32)(12,26,27,36,28,31,13,29)Polycyclic presentation:
2^9|a^2=i,b^2=fi,e^2=fi,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,h^b=hi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426266);