Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469989 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1492 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)(3,8)(4,7)(5,6)(9,10)(11,14)(12,15)(13,16)(17,18,23,36,22,26,21,27)(19,28,35,24,30,43,31,39)(20,25,37,48,33,41,34,42)(29,44,47,38,32,46,45,40);(2,6)(7,8)(9,11,14,10)(12,15,13,16)(17,19,22,30)(18,24,26,39)(20,32,33,29)(21,35,23,31)(25,40,41,38)(27,43,36,28)(34,45,37,47)(42,44,48,46);(1,3,5,4)(2,7,6,8)(10,11)(15,16)(17,20)(18,25)(19,29)(21,34)(22,33)(23,37)(24,38)(26,41)(27,42)(28,44)(30,32)(31,45)(35,47)(36,48)(39,40)(43,46);(9,12,14,13)(10,15,11,16)(18,26)(19,30)(25,41)(27,36)(29,32)(31,35)(42,48)(45,47);(1,4,5,3)(2,7,6,8)(9,13,14,12)(10,15,11,16)(18,26)(20,33)(24,39)(27,36)(28,43)(29,32)(34,37)(45,47)Polycyclic presentation:
2^9|a^2=fgi,b^2=h,c^2=ghi,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dh,f^b=fi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6397987);