Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573635 |
| 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 | |
4528 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)(7,20)(8,26)(9,15)(10,16)(11,19)(12,18)(13,24)(14,29)(17,28)(21,32)(22,36)(23,31)(25,34)(27,33)(30,35);(5,7,12,25)(6,13,18,33)(8,23,29,30)(9,22,21,10)(11,27,28,24)(14,15,26,32)(16,31,36,35)(17,34,19,20);(2,4)(5,8,12,29)(6,14,18,26)(7,21,25,9)(10,28,22,11)(13,31,33,35)(15,34,32,20)(16,19,36,17)(23,24,30,27);(5,9,12,21)(6,15,18,32)(7,22,25,10)(8,27,29,24)(11,23,28,30)(13,26,33,14)(16,20,36,34)(17,35,19,31);(5,10)(6,16)(7,23)(8,28)(9,27)(11,29)(12,22)(13,32)(14,19)(15,33)(17,26)(18,36)(20,31)(21,24)(25,30)(34,35)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480836);