Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437608 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
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 | |
5887 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)(5,6,12,17)(7,18,22,29)(8,23,28,32)(9,15,13,21)(10,20,11,16)(14,26,33,27)(19,30,36,25)(24,31,35,34);(2,4)(5,7,12,22)(6,13,17,9)(8,24,28,35)(10,21,11,15)(14,31,33,34)(16,29,20,18)(19,27,36,26)(23,30,32,25);(5,8)(6,14)(7,19)(9,25)(10,26)(11,27)(12,28)(13,30)(15,24)(16,32)(17,33)(18,34)(20,23)(21,35)(22,36)(29,31);(5,9)(6,15)(7,20)(8,25)(10,29)(11,18)(12,13)(14,24)(16,22)(17,21)(19,23)(26,31)(27,34)(28,30)(32,36)(33,35);(5,10,12,11)(6,16,17,20)(7,15,22,21)(8,26,28,27)(9,29,13,18)(14,32,33,23)(19,24,36,35)(25,31,30,34)Polycyclic presentation:
2^9|a^2=i,b^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,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, 6344808);