Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573643 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,12)(8,18)(9,23)(10,15)(11,16)(13,28)(14,30)(17,24)(19,22)(20,29)(21,35)(25,27)(26,33)(31,34)(32,36);(1,2,3,4)(5,7,11,20)(6,12,16,29)(8,14,22,15)(9,17,28,34)(10,18,30,19)(13,31,23,24)(21,33,36,25)(26,32,27,35);(5,8,11,22)(6,7,16,20)(9,24,28,31)(10,29,30,12)(13,32,23,35)(14,19,15,18)(17,25,34,33)(21,26,36,27);(5,9,11,28)(6,13,16,23)(7,17,20,34)(8,21,22,36)(10,26,30,27)(12,31,29,24)(14,33,15,25)(18,32,19,35);(5,6)(7,8)(9,25)(10,14)(11,16)(12,18)(13,26)(15,30)(17,31)(19,29)(20,22)(21,35)(23,27)(24,34)(28,33)(32,36)Polycyclic presentation:
2^9|b^2=fi,c^2=i,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481194);