Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| 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 | #25388 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1572 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,7)(3,4,8,5)(9,10,12,13)(11,15,16,14)(17,18,21,26)(19,28,30,23)(20,32,33,25)(22,24,35,27)(29,37,42,41)(31,38,44,36)(34,40,45,39)(43,46,48,47);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,29)(21,30)(22,31)(24,36)(25,37)(26,28)(27,38)(32,41)(33,42)(34,43)(35,44)(39,46)(40,47)(45,48);(1,3)(2,4)(5,7)(6,8)(10,13)(14,15)(18,24)(20,33)(22,35)(23,36)(25,40)(26,27)(28,38)(29,42)(31,44)(32,39)(37,47)(41,46);(1,4,6,5)(2,3,7,8)(9,11,12,16)(10,14,13,15)(17,20)(18,25)(19,29)(21,33)(22,34)(23,37)(24,39)(26,32)(27,40)(28,41)(30,42)(31,43)(35,45)(36,46)(38,47)(44,48);(1,5,6,4)(2,3,7,8)(19,30)(20,33)(23,28)(25,32)(31,44)(34,45)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=f,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6872024);