Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6367198 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,12)(4,9)(5,10)(6,13)(7,11)(8,15)(14,16)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,32)(29,31);(1,3,7,14)(2,8,11,5)(4,15,6,10)(9,16,13,12)(17,18)(19,22)(20,24)(21,23)(25,26)(27,31)(28,30)(29,32);(1,4,7,6)(2,9,11,13)(3,10,14,15)(5,16,8,12)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25);(1,5)(2,10)(3,13)(4,16)(6,12)(7,8)(9,14)(11,15);(2,11)(3,14)(9,13)(10,15)(18,23)(22,31)(24,28)(25,32)Polycyclic presentation:
2^9|b^2=h,c^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72306796, #72306825, #72306900, … |
| maximal normal subgroup of | #72306796, #72306825, #72306900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274458);