Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514160 |
| 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 | |
2529 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(5,6)(7,12)(8,19)(9,24)(10,26)(11,16)(13,22)(14,25)(15,28)(17,34)(18,32)(20,36)(21,30)(23,33)(27,31)(29,35);(1,2,4,3)(5,7,11,9)(6,12,16,24)(8,20,23,35)(10,25,28,18)(13,30,31,34)(14,15,32,26)(17,22,21,27)(19,36,33,29);(5,8)(6,13)(7,17)(9,21)(10,27)(11,23)(12,29)(14,30)(15,33)(16,31)(18,20)(19,26)(22,28)(24,36)(25,35)(32,34);(5,9,11,7)(6,14,16,32)(8,21,23,17)(10,18,28,25)(12,26,24,15)(13,30,31,34)(19,36,33,29)(20,22,35,27);(1,3,4,2)Polycyclic presentation:
2^9|b^2=fi,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=cgi,d^b=dh,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, 7425991);