Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,25)(20,26)(21,27)(22,39)(23,29)(24,30)(28,32)(31,40)(33,41)(34,42)(35,43)(36,47)(37,44)(38,45)(46,48);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,24,33)(18,25,30,41)(20,34,38,36)(21,32,23,22)(26,42,45,47)(27,28,29,39)(31,35,46,37)(40,43,48,44);(10,14)(13,15)(17,20,21,35,24,38,23,37)(18,26,27,43,30,45,29,44)(19,31,22,36,33,46,32,34)(25,40,39,47,41,48,28,42);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,24,23)(18,27,30,29)(19,32,33,22)(20,35,38,37)(25,28,41,39)(26,43,45,44)(31,34,46,36)(40,42,48,47);(2,5)(6,8)(11,16)(13,15)(17,22,23,19,24,32,21,33)(18,28,29,41,30,39,27,25)(20,36,37,31,38,34,35,46)(26,42,44,48,45,47,43,40)Polycyclic presentation:
2^9|b^2=i,c^2=hi,d^2=fgi,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6730655);