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 | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1599 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)(19,25)(20,32)(21,37)(22,28)(23,40)(24,29)(26,35)(27,31)(30,46)(33,45)(34,44)(36,48)(38,41)(39,42)(43,47);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,24,31)(18,25,29,27)(20,33,36,38)(21,23,39,22)(26,43,44,46)(28,37,40,42)(30,35,47,34)(32,45,48,41);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,20,23,35,24,36,22,34)(18,26,28,32,29,44,40,48)(19,30,21,38,31,47,39,33)(25,41,42,46,27,45,37,43);(9,11,12,16)(10,14,13,15)(17,21,23,31,24,39,22,19)(18,27,28,37,29,25,40,42)(20,33,35,30,36,38,34,47)(26,41,32,46,44,45,48,43);(1,4,5,3)(2,7,6,8)(17,22,24,23)(18,28,29,40)(19,21,31,39)(20,34,36,35)(25,37,27,42)(26,32,44,48)(30,38,47,33)(41,43,45,46)Polycyclic presentation:
2^9|b^2=i,c^2=fh,d^2=g,e^2=fi,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,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504635);