Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6636852 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1434 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,24,30,39)(20,32,34,40)(21,26,22,36)(25,35,41,33)(28,42,44,47)(29,38,31,46)(37,45,48,43);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,28)(21,29)(22,31)(23,30)(25,37)(26,38)(27,39)(32,42)(33,43)(34,44)(35,45)(36,46)(40,47)(41,48);(11,15)(13,16)(17,20,21,35,23,34,22,33)(18,25,36,40,27,41,26,32)(19,28,29,45,30,44,31,43)(24,37,46,47,39,48,38,42);(2,6)(3,8)(11,15)(13,16)(17,21,23,22)(18,26,27,36)(19,29,30,31)(20,33,34,35)(24,38,39,46)(25,40,41,32)(28,43,44,45)(37,47,48,42);(2,6)(5,7)(10,14)(13,16)(19,30)(24,39)(28,44)(29,31)(37,48)(38,46)(42,47)(43,45)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,d^c=di56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6544369);