Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1239 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,6,4,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,24)(19,27,30,23)(20,32,34,25)(22,26,36,39)(28,37,31,42)(29,38,33,40)(35,45,46,41)(43,48,44,47);(1,2)(3,6)(4,8)(5,7)(17,19,21,30)(18,23,24,27)(20,33,34,29)(22,28,36,31)(25,40,32,38)(26,42,39,37)(35,44,46,43)(41,47,45,48);(1,3,5,4)(2,6,7,8)(10,13)(14,16)(18,24)(19,28)(22,36)(23,37)(25,32)(27,42)(29,43)(30,31)(33,44)(35,46)(38,47)(40,48);(2,7)(6,8)(10,13)(14,16)(17,20)(18,25)(19,29)(21,34)(22,35)(23,38)(24,32)(26,41)(27,40)(28,43)(30,33)(31,44)(36,46)(37,47)(39,45)(42,48);(1,4,5,3)(2,8,7,6)(9,11,12,15)(10,14,13,16)(18,24)(20,34)(23,27)(26,39)(29,33)(35,46)(37,42)(43,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,h^b=hi,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7420352);