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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1753 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)(3,5,7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,24,22,28,21,27)(19,29,32,39,30,25,33,37)(20,26,36,38,35,42,34,41)(31,43,45,48,44,40,46,47);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,14)(13,16)(18,24)(19,30)(20,34)(21,23)(25,39)(27,28)(29,37)(31,45)(35,36)(38,41)(40,43)(44,46);(2,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,25)(20,31)(21,33)(22,30)(23,32)(24,37)(26,40)(27,39)(28,29)(34,46)(35,44)(36,45)(38,47)(41,48)(42,43);(11,16)(13,15)(17,20,23,36,22,35,21,34)(18,26,24,38,28,42,27,41)(19,31,32,45,30,44,33,46)(25,40,37,47,29,43,39,48);(10,14)(13,15)(17,21,22,23)(18,24,28,27)(19,32,30,33)(20,34,35,36)(25,39,29,37)(26,38,42,41)(31,45,44,46)(40,48,43,47)Polycyclic presentation:
2^9|a^2=fg,b^2=gi,d^2=fi,e^2=i,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, 6926349);