Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| 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 | #23580 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1465 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)(3,7)(4,6)(5,8)(17,18,22,27,21,26,35,24)(19,28,31,37,30,23,36,39)(20,32,34,40,33,25,41,42)(29,38,45,48,44,43,46,47);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19)(18,23)(20,29)(21,30)(22,31)(24,37)(25,38)(26,28)(27,39)(32,43)(33,44)(34,45)(35,36)(40,47)(41,46)(42,48);(3,8)(5,7)(9,10)(11,14)(12,13)(15,16)(18,24)(20,33)(22,35)(23,37)(25,42)(26,27)(28,39)(29,44)(31,36)(32,40)(38,48)(43,47);(2,6)(5,7)(10,13)(14,16)(17,20)(18,25)(19,29)(21,33)(22,34)(23,38)(24,40)(26,32)(27,42)(28,43)(30,44)(31,45)(35,41)(36,46)(37,47)(39,48);(9,11,12,15)(10,14,13,16)(19,30)(20,33)(23,28)(25,32)(31,36)(34,41)(37,39)(40,42)Polycyclic presentation:
2^9|a^2=g,b^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,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, 6873550);