Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439025 |
| 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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6)(7,12)(8,20)(9,24)(10,14)(11,15)(13,22)(16,30)(17,25)(18,28)(19,29)(21,35)(23,31)(26,32)(27,34)(33,36);(2,3)(5,7)(6,12)(8,21)(9,19)(10,25)(11,18)(13,30)(14,17)(15,28)(16,22)(20,35)(23,34)(24,29)(26,33)(27,31)(32,36);(1,2,4,3)(5,8,10,26,11,23,9,22)(6,13,24,31,15,32,14,20)(7,16,25,21,18,33,19,34)(12,27,29,36,28,35,17,30);(1,3,4,2)(5,6)(7,17)(8,13)(9,14)(10,24)(11,15)(12,25)(16,30)(18,29)(19,28)(20,22)(21,27)(23,32)(26,31)(33,36)(34,35);(7,18)(12,28)(16,33)(17,29)(19,25)(21,34)(27,35)(30,36)Polycyclic presentation:
2^9|a^2=f,c^2=fgi,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351496);