Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574070 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 | |
3014 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)(7,15)(8,13)(9,14)(10,17)(11,18)(12,20)(16,22)(19,35)(21,30)(23,31)(24,29)(25,32)(26,36)(27,33)(28,34);(5,7,10,20,11,8,22,21)(6,12,16,15,18,30,17,13)(9,19,26,24,28,23,27,25)(14,29,33,35,34,32,36,31);(1,2,3,4)(5,8)(6,13)(7,11)(9,24)(10,21)(12,16)(14,29)(15,18)(17,30)(19,27)(20,22)(23,26)(25,28)(31,36)(32,34)(33,35);(5,9)(6,14)(7,19)(8,23)(10,26)(11,28)(12,29)(13,31)(15,35)(16,33)(17,36)(18,34)(20,24)(21,25)(22,27)(30,32);(5,7)(6,15)(8,11)(9,25)(10,20)(12,17)(13,18)(14,32)(16,30)(19,26)(21,22)(23,27)(24,28)(29,34)(31,33)(35,36)Polycyclic presentation:
2^9|b^2=f,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6490386);