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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1516 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)(3,4)(5,6)(7,8)(17,18)(19,23)(20,25)(21,34)(22,27)(24,33)(26,36)(28,44)(29,38)(30,39)(31,48)(32,43)(35,42)(37,45)(40,47)(41,46);(2,6)(4,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,27,39)(20,29,32,28)(21,31,36,47)(24,41,42,37)(25,38,43,44)(26,40,34,48)(33,46,35,45);(1,3)(2,7)(4,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,20,22,32)(18,24,27,42)(19,28,30,29)(21,35,36,33)(23,37,39,41)(25,26,43,34)(31,46,47,45)(38,48,44,40);(2,6)(3,8)(11,16)(13,15)(17,20)(18,25)(19,29)(21,33)(22,32)(23,38)(24,34)(26,42)(27,43)(28,30)(31,46)(35,36)(37,40)(39,44)(41,48)(45,47);(1,4,5,7)(2,3,6,8)(10,14)(13,15)(19,30)(20,32)(23,39)(24,42)(25,43)(31,47)(33,35)(40,48)Polycyclic presentation:
2^9|b^2=i,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,e^c=ei,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, 6796465);