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 | |
1624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,7)(4,5)(8,10)(9,15)(11,16)(12,14)(19,26)(20,21)(23,30)(24,28)(27,29)(31,32);(1,2,6,8)(3,11,13,15)(4,7,5,10)(9,14,16,12)(17,18)(19,23)(20,28)(21,24)(22,25)(26,30)(27,32)(29,31);(2,8)(7,10)(9,16)(11,15)(17,19,22,27)(18,23,25,32)(20,26,21,29)(24,31,28,30);(1,3)(2,9)(4,12)(5,14)(6,13)(7,15)(8,16)(10,11);(1,4,6,5)(2,10,8,7)(3,12,13,14)(9,11,16,15)(17,20,22,21)(18,24,25,28)(19,26,27,29)(23,31,32,30)Polycyclic presentation:
2^9|b^2=h,c^2=i,e^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 | #72306850, #72306854, #72306878, … |
| maximal normal subgroup of | #72306850, #72306854, #72306878, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274654);