Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439029 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 | |
4441 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)(3,4)(5,6,12,18)(7,14,20,29)(8,22,21,13)(9,15,27,33)(10,17,11,16)(19,30,34,36)(23,35,24,28)(25,32,26,31);(2,4)(5,7)(6,13)(8,10)(9,24)(11,21)(12,20)(14,17)(15,30)(16,29)(18,22)(19,26)(23,27)(25,34)(28,32)(31,35)(33,36);(5,8,11,7,12,21,10,20)(6,14,17,22,18,29,16,13)(9,23,25,19,27,24,26,34)(15,30,31,35,33,36,32,28);(5,9)(6,15)(7,19)(8,23)(10,26)(11,25)(12,27)(13,28)(14,30)(16,32)(17,31)(18,33)(20,34)(21,24)(22,35)(29,36);(7,20)(8,21)(13,22)(14,29)(19,34)(23,24)(28,35)(30,36)Polycyclic presentation:
2^9|a^2=i,c^2=g,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349914);