Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357777 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2047 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,9)(3,10,12,16)(4,8,5,14)(7,11,15,13)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,31)(30,32);(1,3,5,13,6,12,4,11)(2,7,8,10,9,15,14,16)(18,22)(23,30)(24,29)(25,31);(17,19,21,27,20,26,28,32)(18,23,24,25,22,30,29,31);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,10,15,16)(19,26)(23,30)(25,31)(27,32);(3,12)(7,15)(10,16)(11,13)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=g,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,e^b=eh,d^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72219655, #72306705, #72306708, … |
| maximal normal subgroup of | #72219655, #72306705, #72306708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264987);