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 | |
2543 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)(5,6,11,15)(7,12,18,32)(8,20,13,28)(9,23,25,22)(10,26,29,14)(16,24,35,17)(19,34,27,33)(21,30,31,36);(1,2,4,3)(5,7)(6,12)(8,21)(9,24)(10,27)(11,18)(13,31)(14,34)(15,32)(16,22)(17,25)(19,29)(20,30)(23,35)(26,33)(28,36);(5,8)(6,9)(7,16)(10,23)(11,13)(12,30)(14,28)(15,25)(17,33)(18,35)(19,31)(20,26)(21,27)(22,29)(24,34)(32,36);(5,9,11,25)(6,13,15,8)(7,17,18,24)(10,28,29,20)(12,31,32,21)(14,22,26,23)(16,33,35,34)(19,36,27,30);(1,3,4,2)(7,18)(12,32)(16,35)(17,24)(19,27)(21,31)(30,36)(33,34)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=i,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, 7426001);