Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439035 |
| 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 | |
2749 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)(7,11)(8,21)(9,14)(10,15)(12,24)(13,26)(16,33)(17,18)(19,28)(20,29)(22,27)(23,32)(25,31)(30,34)(35,36);(2,3)(5,7)(6,11)(8,22)(9,18)(10,20)(12,30)(13,28)(14,17)(15,29)(16,25)(19,26)(21,27)(23,35)(24,34)(31,33)(32,36);(1,2,4,3)(5,8,9,24,10,23,26,25)(6,12,13,21,15,31,14,32)(7,16,18,22,20,34,19,35)(11,27,28,33,29,36,17,30);(1,3,4,2)(5,6)(7,17)(8,12)(9,13)(10,15)(11,18)(14,26)(16,30)(19,29)(20,28)(21,24)(22,27)(23,31)(25,32)(33,34)(35,36);(8,23)(12,31)(16,34)(21,32)(22,35)(24,25)(27,36)(30,33)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,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, 6351501);