Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| 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 | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1877 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,4,7,6)(9,10)(11,15)(12,14)(13,16)(17,18,20,23)(19,25,27,22)(21,24,29,32)(26,31,28,30);(1,3,5,7)(2,6,8,4)(9,11)(10,13)(12,16)(14,15)(17,19,20,27)(18,22,23,25)(21,26,29,28)(24,30,32,31);(1,4,5,6)(2,7,8,3)(10,14)(13,15)(19,26)(21,29)(22,30)(24,32)(25,31)(27,28);(2,8)(3,7)(11,16)(13,15)(18,23)(19,27)(24,32)(26,28);(19,27)(22,25)(26,28)(30,31)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72316784, #72316786, #72316791, … |
| maximal normal subgroup of | #72316784, #72316786, #72316791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7417021);