Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519083 |
| 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 | |
2500 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)(7,13,19,28)(8,20,22,35)(9,25,10,15)(12,26,29,18)(14,24,32,23)(17,31,21,27)(30,36,34,33);(1,2)(3,4)(5,7,10,26,11,19,9,18)(6,12,25,28,16,29,15,13)(8,21,24,33,22,17,23,36)(14,31,35,30,32,27,20,34);(2,4)(8,22)(14,32)(17,21)(20,35)(23,24)(27,31)(30,34)(33,36);(2,4)(5,7)(6,13)(8,17)(9,18)(10,26)(11,19)(12,25)(14,30)(15,29)(16,28)(20,31)(21,22)(23,33)(24,36)(27,35)(32,34);(5,8,9,23,11,22,10,24)(6,14,15,20,16,32,25,35)(7,17,18,33,19,21,26,36)(12,27,13,30,29,31,28,34)Polycyclic presentation:
2^9|a^2=i,b^2=fi,e^2=f,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,e^b=ei,h^b=hi,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426290);