Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1620 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)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,26)(20,24)(21,23)(22,31)(25,29)(27,32)(28,30);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,29)(18,25,21,26)(20,30,31,27)(22,28,24,32);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,23,31)(18,24,21,22)(19,27,29,30)(25,28,26,32);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,21,23,18)(19,27,29,30)(20,22,31,24)(25,32,26,28);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,22,23,24)(18,20,21,31)(19,28,29,32)(25,30,26,27)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=fg,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7496536);