Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632048 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35149 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2293 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)(21,26,22,35)(25,36,41,33)(28,42,44,47)(29,38,31,45)(37,46,48,43);(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)(18,25)(19,28)(21,33)(22,36)(23,34)(24,37)(26,40)(27,41)(29,43)(30,44)(31,46)(32,35)(38,47)(39,48)(42,45);(2,6)(3,8)(11,15)(13,16)(17,21,23,22)(18,26,27,35)(19,29,30,31)(20,33,34,36)(24,38,39,45)(25,40,41,32)(28,43,44,46)(37,47,48,42);(2,6)(5,7)(10,14)(13,16)(19,30)(24,39)(28,44)(29,31)(37,48)(38,45)(42,47)(43,46)Polycyclic presentation:
2^9|a^2=i,b^2=f,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,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6539388);