Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(11,15)(14,16)(17,18,23,38,22,28,21,27)(19,29,31,24,33,25,32,41)(20,26,39,48,37,45,36,44)(30,43,46,40,35,42,34,47);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(17,19)(18,24)(20,34)(21,31)(22,33)(23,32)(25,27)(26,43)(28,41)(29,38)(30,36)(35,39)(37,46)(40,44)(42,45)(47,48);(2,6)(5,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,35)(21,32)(22,33)(23,31)(24,27)(26,43)(28,29)(30,37)(34,39)(36,46)(38,41)(40,48)(42,45)(44,47);(3,8)(5,7)(10,13)(14,16)(17,20,23,39,22,37,21,36)(18,26,38,48,28,45,27,44)(19,30,31,46,33,35,32,34)(24,40,25,42,41,47,29,43);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,27,28,38)(19,31,33,32)(20,36,37,39)(24,25,41,29)(26,44,45,48)(30,46,35,34)(40,42,47,43)Polycyclic presentation:
2^9|a^2=fi,d^2=fg,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7067788);