Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433057 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1351 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)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,24,30)(19,26,33,41)(20,25,32,42)(21,35,37,27)(22,29,23,28)(31,47,36,43)(34,44,48,40)(38,45,39,46);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,20,24,33,23,32)(18,25,28,26,30,42,29,41)(21,36,38,48,37,31,39,34)(27,44,45,47,35,40,46,43);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,20,22,33,24,32,23,19)(18,26,28,42,30,41,29,25)(21,34,38,36,37,48,39,31)(27,43,45,44,35,47,46,40);(10,14)(13,16)(17,21,24,37)(18,27,30,35)(19,31,33,36)(20,34,32,48)(22,38,23,39)(25,40,42,44)(26,43,41,47)(28,45,29,46);(1,4,5,3)(2,6,8,7)(21,37)(27,35)(31,36)(34,48)(38,39)(40,44)(43,47)(45,46)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,d^2=i,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341825);