Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| 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 | #2540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18,23,28)(19,30,32,25)(20,33,37,43)(21,39,40,45)(22,29,24,27)(26,36,44,38)(31,42,47,48)(34,41,35,46);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,32)(18,25,28,30)(20,34,37,35)(21,24,40,22)(26,42,44,48)(27,39,29,45)(31,38,47,36)(33,46,43,41);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,20,22,36,23,37,24,38)(18,26,27,43,28,44,29,33)(19,31,40,34,32,47,21,35)(25,41,45,42,30,46,39,48);(9,11,12,16)(10,14,13,15)(17,21,24,32,23,40,22,19)(18,25,29,39,28,30,27,45)(20,35,38,47,37,34,36,31)(26,41,33,48,44,46,43,42);(1,4,5,3)(2,7,6,8)(17,22,23,24)(18,27,28,29)(19,21,32,40)(20,36,37,38)(25,39,30,45)(26,43,44,33)(31,35,47,34)(41,48,46,42)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fh,d^2=g,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504596);