Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1538 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)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,28)(19,23,33,40)(20,34,37,44)(21,26,29,27)(24,31,39,32)(25,42,45,36)(30,43,47,41)(35,38,46,48);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,35)(21,31)(22,33)(24,27)(25,43)(26,39)(28,40)(29,32)(30,36)(34,38)(37,46)(41,45)(42,47)(44,48);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,19,29,31,22,33,21,32)(18,24,26,40,28,39,27,23)(20,30,42,46,37,47,36,35)(25,41,44,48,45,43,34,38);(1,4,5,3)(2,6,8,7)(9,11,12,16)(10,14,13,15)(17,20,21,36,22,37,29,42)(18,25,27,34,28,45,26,44)(19,30,32,35,33,47,31,46)(23,38,39,43,40,48,24,41);(2,8)(6,7)(17,21,22,29)(18,26,28,27)(19,31,33,32)(20,36,37,42)(23,39,40,24)(25,44,45,34)(30,46,47,35)(38,43,48,41)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fhi,d^2=fg,e^2=i,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=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7503310);