Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470003 |
| 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 | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,29,33,44)(20,25,35,42)(21,26,36,43)(22,37,34,27)(24,38,41,32)(30,40,46,48)(31,45,47,39);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,30)(21,31)(22,38)(23,33)(25,39)(26,40)(27,44)(28,41)(29,37)(32,34)(35,46)(36,47)(42,45)(43,48);(2,6)(5,8)(11,15)(13,16)(17,20,23,35)(18,25,28,42)(19,30,33,46)(21,34,36,22)(24,39,41,45)(26,27,43,37)(29,40,44,48)(31,32,47,38);(11,15)(13,16)(17,21)(18,26)(19,31)(20,34)(22,35)(23,36)(24,40)(25,27)(28,43)(29,45)(30,32)(33,47)(37,42)(38,46)(39,44)(41,48);(10,14)(13,16)(20,35)(21,36)(25,42)(26,43)(30,46)(31,47)(39,45)(40,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6380028);