Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470355 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1765 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)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,25)(19,26,24,21)(20,23,28,30)(27,32,31,29);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,18,22,24,26,25)(20,27,29,23,28,31,32,30);(1,3,4,8)(2,6,5,7)(11,15)(13,16)(17,20)(18,23)(19,27)(21,29)(22,28)(24,31)(25,30)(26,32);(11,15)(13,16)(20,28)(23,30)(27,31)(29,32);(10,14)(13,16)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6382662);