Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439025 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 | |
4598 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)(6,12)(7,17)(9,10)(13,26)(14,22)(15,16)(19,31)(20,21)(23,24)(27,35)(29,30)(33,34);(2,4)(5,6)(7,18)(8,22)(9,16)(10,12)(11,15)(13,20)(14,24)(17,26)(19,27)(21,28)(23,29)(25,30)(31,32)(33,36)(34,35);(5,7,9,20,11,21,10,17)(6,13,16,28,15,26,12,18)(8,19,24,33,25,34,23,31)(14,27,30,36,29,35,22,32);(5,8)(6,14)(7,19)(9,24)(10,23)(11,25)(12,22)(13,27)(15,29)(16,30)(17,31)(18,32)(20,33)(21,34)(26,35)(28,36);(6,15)(12,16)(13,26)(14,29)(18,28)(22,30)(27,35)(32,36)Polycyclic presentation:
2^9|c^2=gi,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349907);