Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432197 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#397 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2769 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,8,7,12,5,11)(3,13,16,24,14,21,20,9)(4,10,19,22,17,25,18,26)(15,27,29,32,28,31,30,23);(2,8)(3,14)(5,6)(9,21)(10,22)(11,12)(13,24)(15,28)(18,19)(23,31)(25,26)(27,32);(1,3,5,16,7,14,6,20)(2,9,11,13,12,24,8,21)(4,15,18,29,17,28,19,30)(10,23,26,27,25,32,22,31);(1,4)(2,10)(3,15)(5,18)(6,19)(7,17)(8,22)(9,23)(11,26)(12,25)(13,27)(14,28)(16,29)(20,30)(21,31)(24,32);(4,17)(10,25)(15,28)(18,19)(22,26)(23,32)(27,31)(29,30)Polycyclic presentation:
2^9|a^2=g,c^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,f^b=fi,g^b=gi,g^c=gi,h^c=hi,e^d=ei89 conjugacy classes — view the full character table.
| maximal subgroup of | #72306905, #72307021, #72307052, … |
| maximal normal subgroup of | #72306905, #72307021, #72307052, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339393);