Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6349099 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2945 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)(3,11)(4,10)(5,8)(6,9)(7,13)(12,16)(14,15)(17,18)(19,25)(20,28)(21,23)(22,27)(24,29)(26,32)(30,31);(1,3,6,14)(2,7,9,16)(4,12,5,13)(8,15,10,11)(20,29)(24,28)(25,32)(27,30);(2,8)(4,5)(7,15)(9,10)(11,16)(12,13)(17,19)(18,22)(20,30)(21,26)(23,31)(24,32)(25,28)(27,29);(1,4,6,5)(2,8,9,10)(3,12,14,13)(7,15,16,11)(18,23)(22,31)(24,28)(25,32);(2,9)(7,16)(8,10)(11,15)Polycyclic presentation:
2^9|b^2=h,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72306697, #72306721, #72307028, … |
| maximal normal subgroup of | #72306697, #72306721, #72307028, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6256298);