Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500186 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,5)(4,6)(7,8)(17,18,22,27,23,28,20,25)(19,29,30,39,32,42,37,24)(21,34,36,40,35,26,33,41)(31,45,44,38,46,43,48,47);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,30)(21,31)(22,37)(23,32)(25,29)(26,38)(27,42)(28,39)(33,44)(34,47)(35,46)(36,48)(40,43)(41,45);(2,6)(5,8)(10,13)(11,15)(17,20,23,22)(18,25,28,27)(19,30,32,37)(21,33,35,36)(24,29,39,42)(26,40,34,41)(31,44,46,48)(38,43,47,45);(9,11,12,15)(10,14,13,16)(17,21)(18,26)(19,31)(20,33)(22,36)(23,35)(24,38)(25,40)(27,41)(28,34)(29,43)(30,44)(32,46)(37,48)(39,47)(42,45);(10,13)(14,16)(21,35)(26,34)(31,46)(33,36)(38,47)(40,41)(43,45)(44,48)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,d^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6426394);