Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | #39334 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1830 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)(3,7)(4,6)(5,8)(17,18)(19,24)(20,29)(21,26)(22,31)(23,27)(25,28)(30,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,24,27,31)(20,28,21,30)(25,26,32,29);(3,8)(5,7)(9,11,12,16)(10,13,14,15)(17,20,23,21)(18,25,27,32)(19,28,22,30)(24,26,31,29);(2,6)(5,7)(17,21,23,20)(18,26,27,29)(19,28,22,30)(24,25,31,32);(10,14)(13,15)(20,21)(25,32)(26,29)(28,30)Polycyclic presentation:
2^9|b^2=i,c^2=fhi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72257482, #72257512, #72264300 |
| maximal normal subgroup of | #72257482, #72257512, #72264300 |
| maximal quotient of | #72264803 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6660494);