Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6362950 |
| 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 |
| prime radical | #61465 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1652 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)(3,10)(4,5)(6,16)(7,9)(8,13)(11,12)(14,15)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32);(1,3,7,14)(2,8,9,11)(4,15,16,10)(5,13,6,12)(17,19,21,27)(18,22,24,30)(20,26,28,32)(23,29,25,31);(18,23)(20,28)(22,29)(24,25)(26,32)(30,31);(1,4,7,16)(2,5,9,6)(3,11,14,8)(10,12,15,13);(1,5,7,6)(2,4,9,16)(3,12,14,13)(8,10,11,15)(18,24)(22,30)(23,25)(29,31)Polycyclic presentation:
2^9|b^2=hi,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=df,e^b=eh,f^b=fh,g^c=gi,f^d=fh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72306533, #72306686, #72306694, … |
| maximal normal subgroup of | #72306533, #72306686, #72306694, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6270198);