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 | |
2519 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)(7,17)(8,20)(9,12)(10,15)(11,27)(13,24)(14,25)(16,26)(18,31)(19,30)(21,29)(22,32)(23,34)(28,33)(35,36);(1,2,4,3)(5,7,10,9)(6,12,15,17)(8,21,23,36)(11,25,26,19)(13,31,33,32)(14,27,30,16)(18,24,22,28)(20,35,34,29);(5,8)(6,13)(7,18)(9,22)(10,23)(11,28)(12,29)(14,32)(15,33)(16,34)(17,35)(19,21)(20,27)(24,26)(25,36)(30,31);(5,9,10,7)(6,14,15,30)(8,22,23,18)(11,19,26,25)(12,16,17,27)(13,32,33,31)(20,29,34,35)(21,24,36,28);(1,3,4,2)(6,15)(12,17)(13,33)(14,30)(16,27)(20,34)(29,35)(31,32)Polycyclic presentation:
2^9|b^2=f,d^2=i,e^2=fi,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, 7425994);