Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514172 |
| 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 | |
2597 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(5,6)(7,11)(8,19)(9,13)(10,15)(12,21)(14,25)(16,30)(17,28)(18,27)(20,26)(22,32)(23,29)(24,31)(33,36)(34,35);(1,2,4,3)(5,7,10,17)(6,11,15,28)(8,20,22,34)(9,23,25,18)(12,30,31,36)(13,29,14,27)(16,24,33,21)(19,26,32,35);(5,8)(6,12)(7,16)(9,24)(10,22)(11,26)(13,19)(14,32)(15,31)(17,33)(18,34)(20,23)(21,25)(27,30)(28,35)(29,36);(6,13)(9,25)(11,27)(12,19)(14,15)(18,23)(20,34)(21,24)(26,30)(28,29)(31,32)(35,36);(1,3,4,2)(7,17)(11,28)(16,33)(18,23)(20,34)(26,35)(27,29)(30,36)Polycyclic presentation:
2^9|b^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7425998);