Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6346594 |
| 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 | C22×C4 | C25 |
| prime radical | C2×#3490 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,10)(3,11,14,27)(4,15,17,29)(5,8,19,23)(7,20,22,13)(9,24,26,18)(12,25,28,32)(16,21,30,31);(1,3)(2,7)(4,16)(5,12)(6,14)(8,21)(9,25)(10,22)(11,15)(13,18)(17,30)(19,28)(20,24)(23,31)(26,32)(27,29);(4,17)(9,26)(11,27)(13,20)(15,29)(16,30)(18,24)(25,32);(3,12)(7,21)(11,25)(13,16)(14,28)(20,30)(22,31)(27,32);(2,8)(7,21)(9,15)(10,23)(11,25)(22,31)(26,29)(27,32)Polycyclic presentation:
2^9|a^2=i,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,d^b=dh,f^b=fh,g^b=gi,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72241080, #72241088, #72241093, … |
| maximal normal subgroup of | #72241080, #72241088, #72241093, … |
| maximal quotient of | #72257810 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253798);