Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1305 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)(11,15)(12,14)(13,16)(17,18,23,28)(19,30,32,24)(20,25,33,42)(21,29,34,26)(22,37,39,27)(31,40,47,46)(35,44,38,41)(36,43,48,45);(9,11)(10,13)(12,16)(14,15)(17,19,23,32)(18,24,28,30)(20,34,33,21)(22,35,39,38)(25,29,42,26)(27,44,37,41)(31,36,47,48)(40,45,46,43);(2,6)(7,8)(10,14)(11,16)(17,20,21,19,23,33,34,32)(18,25,29,30,28,42,26,24)(22,38,36,47,39,35,48,31)(27,44,45,40,37,41,43,46);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,21,23,34)(18,26,28,29)(19,20,32,33)(22,36,39,48)(24,25,30,42)(27,43,37,45)(31,35,47,38)(40,41,46,44);(1,4,5,3)(2,8,6,7)(17,22,34,48,23,39,21,36)(18,27,26,43,28,37,29,45)(19,31,20,35,32,47,33,38)(24,40,42,44,30,46,25,41)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fhi,d^2=fg,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6749326);