Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35761 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1910 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,18,21,25)(19,26,29,23)(20,22,30,24)(27,31,28,32);(10,14)(13,16)(18,22)(19,27)(20,30)(23,26)(24,25)(28,29);(1,3,4,7)(2,6,5,8)(17,19)(18,23)(20,28)(21,29)(22,31)(24,32)(25,26)(27,30);(18,22)(19,28)(20,30)(24,25)(27,29)(31,32)Polycyclic presentation:
2^9|b^2=gi,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72256829, #72256835, #72256838, … |
| maximal normal subgroup of | #72256829, #72256835, #72256838, … |
| maximal quotient of | #72264355 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6486862);