Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | C2×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1296 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18)(19,30)(20,26)(21,35)(22,28)(23,29)(24,27)(25,33)(31,39)(32,45)(34,41)(36,48)(37,44)(38,42)(40,47)(43,46);(9,11)(10,13)(12,16)(14,15)(17,19,23,33)(18,25,29,30)(20,21,31,24)(22,34,37,46)(26,27,39,35)(28,43,44,41)(32,38,47,36)(40,42,45,48);(2,6)(7,8)(11,16)(13,15)(17,20,24,19,23,31,21,33)(18,26,27,30,29,39,35,25)(22,34,38,32,37,46,36,47)(28,41,42,45,44,43,48,40);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,21,23,24)(18,27,29,35)(19,31,33,20)(22,36,37,38)(25,39,30,26)(28,42,44,48)(32,46,47,34)(40,43,45,41);(1,4,5,3)(2,8,6,7)(17,22,21,36,23,37,24,38)(18,28,35,48,29,44,27,42)(19,32,20,34,33,47,31,46)(25,40,39,43,30,45,26,41)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fhi,d^2=gi,e^2=fgh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6751198);