Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632020 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #34963 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1884 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10,12,14)(11,15,16,13)(17,18)(19,26)(20,24)(21,23)(22,27)(25,29)(28,31)(30,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,22)(20,27)(21,28)(23,30)(24,26)(25,31)(29,32);(11,16)(13,15)(18,23)(21,29)(22,30)(24,25)(26,31)(28,32);(2,5)(6,7)(10,14)(13,15)(18,24)(19,27)(23,25)(28,32);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^9|a^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efhi,g^a=gi,c^b=ch,d^b=dfh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72316732, #72316739, #72316762, … |
| maximal normal subgroup of | #72316732, #72316739, #72316762, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6560215);