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 | #45265 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1557 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,23,29)(19,31,21,25)(20,34,37,45)(22,30,24,28)(26,36,46,38)(27,33,42,40)(32,43,39,44)(35,41,48,47);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,35)(21,23)(22,33)(24,40)(26,43)(27,30)(28,42)(29,31)(32,36)(34,47)(37,48)(38,39)(41,45)(44,46);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,20,22,36,23,37,24,38)(18,26,28,45,29,46,30,34)(19,32,40,35,21,39,33,48)(25,41,27,43,31,47,42,44);(9,11,12,16)(10,14,13,15)(17,21,22,33,23,19,24,40)(18,27,28,31,29,42,30,25)(20,32,36,35,37,39,38,48)(26,44,45,41,46,43,34,47);(2,5)(6,8)(17,22,23,24)(18,28,29,30)(19,33,21,40)(20,36,37,38)(25,42,31,27)(26,45,46,34)(32,48,39,35)(41,44,47,43)Polycyclic presentation:
2^9|a^2=i,c^2=fh,d^2=gi,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=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, 7502396);