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 | |
1760 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)(19,25,31,36,29,40,32,38)(20,26,35,37,34,42,33,41)(30,39,44,46,43,48,45,47);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,14)(13,16)(18,24)(19,29)(20,33)(21,23)(25,38)(27,28)(30,44)(34,35)(36,40)(37,41)(39,48)(43,45);(2,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,25)(20,30)(21,32)(22,29)(23,31)(24,36)(26,39)(27,38)(28,40)(33,45)(34,43)(35,44)(37,46)(41,47)(42,48);(11,16)(13,15)(17,20,21,33,22,34,23,35)(18,26,27,41,28,42,24,37)(19,30,32,45,29,43,31,44)(25,39,38,47,40,48,36,46);(10,14)(13,15)(17,21,22,23)(18,24,28,27)(19,31,29,32)(20,33,34,35)(25,38,40,36)(26,37,42,41)(30,44,43,45)(39,47,48,46)Polycyclic presentation:
2^9|a^2=fgi,b^2=g,d^2=f,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, 6926344);