Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | #39357 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1763 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,4)(17,18,22,27)(19,28,29,40)(20,24,33,41)(21,25,30,26)(23,37,38,32)(31,45,34,48)(35,43,36,44)(39,46,42,47);(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,14)(13,16)(19,29)(20,33)(23,38)(24,41)(28,40)(32,37)(35,36)(43,44);(1,3)(2,7)(4,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,30,37,22,29,21,32)(18,23,25,40,27,38,26,28)(20,34,35,47,33,31,36,46)(24,42,44,45,41,39,43,48);(1,4,5,3)(2,8,6,7)(11,16)(13,15)(17,20,22,33)(18,24,27,41)(19,31,29,34)(21,36,30,35)(23,39,38,42)(25,44,26,43)(28,45,40,48)(32,47,37,46);(2,6)(7,8)(10,14)(13,15)(17,21,22,30)(18,25,27,26)(19,32,29,37)(20,35,33,36)(23,40,38,28)(24,43,41,44)(31,46,34,47)(39,45,42,48)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fgi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6818684);