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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
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)(19,22)(20,24)(21,27)(23,25)(26,30)(28,29)(31,32);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,18,22,19)(20,28,30,32)(21,25,23,27)(24,31,26,29);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19,22,18)(20,26,30,24)(21,25,23,27)(28,29,32,31);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,21,22,23)(18,25,19,27)(20,29,30,31)(24,32,26,28)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=fg,e^2=fgi,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, 7496541);