Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6354266 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #60798 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1795 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(4,11)(7,12)(8,13)(10,15)(14,16)(17,18)(19,27)(20,26)(21,23)(22,25)(24,28)(29,32)(30,31);(1,2,4,6,5,8,11,13)(3,7,10,12,9,14,15,16)(18,23)(24,31)(25,26)(27,32);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,27)(26,32);(2,8)(6,13)(7,14)(12,16)(17,20,21,22)(18,25,23,26)(19,28,29,30)(24,27,31,32);(2,8)(3,9)(6,13)(10,15)Polycyclic presentation:
2^9|b^2=fi,d^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,d^b=dh,e^b=eh,e^c=eh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72283748, #72306567, #72306689, … |
| maximal normal subgroup of | #72283748, #72306567, #72306689, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261476);