Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573652 |
| 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 | |
2557 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,13)(8,15)(9,16)(10,14)(11,17)(12,18)(19,28)(20,27)(21,29)(22,31)(23,32)(24,30)(25,34)(26,33)(35,36);(1,2,3,4)(5,7)(6,13)(8,10)(9,23)(11,19)(12,21)(14,15)(16,32)(17,28)(18,29)(20,24)(22,25)(26,35)(27,30)(31,34)(33,36);(6,14)(7,19)(8,21)(10,11)(15,28)(16,30)(17,18)(20,23)(22,35)(24,26)(31,32)(33,34);(5,8,12,19)(6,15,18,28)(7,10,21,11)(9,20,25,35)(13,14,29,17)(16,27,34,36)(22,26,23,24)(30,31,33,32);(5,9)(6,16)(7,20)(8,22)(10,24)(11,26)(12,25)(13,27)(14,30)(15,31)(17,33)(18,34)(19,23)(21,35)(28,32)(29,36)Polycyclic presentation:
2^9|b^2=f,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,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=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481182);