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 | #2615 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1490 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)(4,8)(5,6)(17,18,22,27)(19,29,31,24)(20,26,21,34)(23,38,37,28)(25,36,42,33)(30,40,35,43)(32,41,47,45)(39,46,44,48);(2,6)(4,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,30)(21,35)(22,31)(23,32)(25,39)(26,43)(27,29)(28,41)(33,46)(34,40)(36,48)(37,47)(38,45)(42,44);(1,3,5,8)(2,7,6,4)(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,30)(21,22)(23,36)(24,39)(26,38)(27,42)(28,34)(29,44)(31,35)(32,48)(33,37)(40,41)(43,45)(46,47);(2,6)(3,8)(11,16)(13,15)(17,21,22,20)(18,26,27,34)(19,30,31,35)(23,36,37,33)(24,40,29,43)(25,28,42,38)(32,46,47,48)(39,45,44,41);(1,4,5,7)(2,8,6,3)(10,14)(13,15)(20,21)(25,42)(26,34)(30,35)(33,36)(39,44)(40,43)(46,48)Polycyclic presentation:
2^9|a^2=i,c^2=fi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,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, 6797884);