Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6368846 |
| 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 | |
1528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,9)(3,6,10,14)(4,7,12,8)(11,13,16,15)(17,18,22,24)(19,26,28,30)(20,23,29,27)(21,31,32,25);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(17,19,22,28)(18,20,24,29)(21,23,32,27)(25,30,31,26);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20,22,29)(18,19,24,28)(21,26,32,30)(23,25,27,31);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(17,20,22,29)(18,23,24,27)(19,25,28,31)(21,30,32,26);(3,10)(6,14)(11,16)(13,15)(18,24)(20,29)(25,31)(26,30)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72306650, #72306654, #72306686, … |
| maximal normal subgroup of | #72306650, #72306654, #72306686, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6276071);