Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682380 |
| 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 | #24435 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1586 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)(3,7,8,5)(17,18)(19,28)(20,32)(21,26)(22,25)(23,30)(24,35)(27,34)(29,43)(31,39)(33,46)(36,42)(37,47)(38,44)(40,45)(41,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,23,25,28)(20,33,36,29)(21,31,34,44)(24,40,41,37)(26,38,27,39)(32,43,42,46)(35,47,48,45);(2,6)(5,7)(10,13)(14,16)(17,20,34,48,22,36,21,35)(18,24,26,42,25,41,27,32)(19,29,44,47,30,33,31,45)(23,37,38,43,28,40,39,46);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,25)(19,30)(24,41)(26,27)(29,33)(31,44)(32,42)(45,47);(3,8)(5,7)(10,13)(14,16)(17,21,22,34)(18,26,25,27)(19,31,30,44)(20,35,36,48)(23,38,28,39)(24,42,41,32)(29,45,33,47)(37,43,40,46)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fg,d^2=hi,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=efh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6691867);