Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437418 |
| 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 | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1896 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)(9,10,12,14)(11,13,15,16)(17,18)(19,27)(20,24)(21,23)(22,25)(26,30)(28,32)(29,31);(9,11)(10,13)(12,15)(14,16)(18,23)(21,30)(24,31)(25,26)(27,32)(28,29);(2,5)(6,8)(11,15)(13,16)(17,19,22,20)(18,24,25,27)(21,28,30,29)(23,31,26,32);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,20,22,19)(18,24,25,27)(21,29,30,28)(23,31,26,32);(18,25)(19,20)(23,26)(28,29)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6365856);