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 | |
1705 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)(3,9)(4,6)(5,14)(7,13)(8,15)(10,11)(12,16)(17,18)(19,23)(20,25)(21,24)(22,27)(26,30)(28,31)(29,32);(1,3,6,12)(2,7,4,10)(5,11,15,13)(8,16,14,9)(17,19,22,20)(18,23,27,25)(21,28,30,29)(24,31,26,32);(1,4,6,2)(3,10,12,7)(5,14,15,8)(9,13,16,11)(18,24)(21,30)(23,31)(25,32)(26,27)(28,29);(3,11)(5,15)(7,16)(8,14)(9,10)(12,13)(17,20,22,19)(18,25,27,23)(21,29,30,28)(24,32,26,31);(2,4)(3,12)(8,14)(11,13)(19,20)(23,25)(28,29)(31,32)Polycyclic presentation:
2^9|b^2=hi,c^2=h,d^2=i,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 | #72306539, #72306549, #72306568, … |
| maximal normal subgroup of | #72306539, #72306549, #72306568, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6276169);