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 | |
2518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6)(7,11)(8,21)(9,14)(10,15)(12,23)(13,26)(16,33)(17,19)(18,28)(20,29)(22,27)(24,31)(25,32)(30,34)(35,36);(2,4)(5,7)(6,11)(8,22)(9,19)(10,18)(12,30)(13,29)(14,17)(15,28)(16,24)(20,26)(21,27)(23,34)(25,35)(31,33)(32,36);(5,8,9,23,10,25,26,24)(6,12,13,21,15,31,14,32)(7,16,19,22,18,34,20,35)(11,27,29,33,28,36,17,30);(5,6)(7,17)(8,12)(9,13)(10,15)(11,19)(14,26)(16,30)(18,29)(20,28)(21,23)(22,27)(24,32)(25,31)(33,34)(35,36);(7,18)(11,28)(16,34)(17,29)(19,20)(22,35)(27,36)(30,33)Polycyclic presentation:
2^9|a^2=f,c^2=g,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, 6350148);