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 | |
1526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,9)(3,8,11,6)(5,7,13,10)(12,15,16,14)(18,23)(20,21)(24,31)(25,26)(27,32)(28,30);(1,3,4,11)(2,6,9,8)(5,12,13,16)(7,14,10,15)(17,18)(19,27)(20,25)(21,23)(22,26)(24,28)(29,31)(30,32);(2,7)(5,13)(6,14)(8,15)(9,10)(12,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,28)(29,32)(30,31);(1,3)(2,8)(4,11)(5,12)(6,9)(7,15)(10,14)(13,16)(17,19,22,29)(18,24,26,32)(20,30,21,28)(23,31,25,27);(3,11)(6,8)(12,16)(14,15)(17,20,22,21)(18,25,26,23)(19,28,29,30)(24,31,32,27)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,d^2=h,e^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, 6276118);