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 | |
5912 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)(7,18)(8,22)(9,15)(10,29)(11,17)(12,16)(13,21)(14,25)(19,30)(20,28)(23,31)(24,36)(26,33)(27,34)(32,35);(2,4)(5,7)(6,13)(8,23)(9,16)(10,15)(11,21)(12,20)(14,31)(17,28)(18,29)(19,26)(22,30)(24,34)(25,36)(27,32)(33,35);(5,8,12,27)(6,14,16,33)(7,19,20,36)(9,24,13,30)(10,26,11,25)(15,32,21,23)(17,34,29,22)(18,35,28,31);(5,9)(6,15)(7,17)(8,24)(10,28)(11,18)(12,13)(14,32)(16,21)(19,34)(20,29)(22,36)(23,33)(25,35)(26,31)(27,30);(6,16)(7,20)(14,33)(15,21)(17,29)(19,36)(22,34)(23,32)Polycyclic presentation:
2^9|c^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, 6344805);