Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682460 |
| 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 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1660 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18)(19,24)(20,32)(21,26)(22,36)(23,27)(25,35)(28,41)(29,38)(30,39)(31,45)(33,46)(34,40)(37,43)(42,47)(44,48);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,28)(21,30)(22,31)(23,29)(25,37)(26,39)(27,38)(32,41)(33,44)(34,42)(35,43)(36,45)(40,47)(46,48);(11,15)(13,16)(17,20,22,35,23,33,21,34)(18,25,26,32,27,40,36,46)(19,28,31,43,29,44,30,42)(24,37,39,41,38,47,45,48);(2,6)(5,7)(19,29)(20,33)(24,38)(25,40)(30,31)(32,46)(34,35)(39,45);(3,8)(5,7)(11,15)(13,16)(17,21,23,22)(18,26,27,36)(19,30,29,31)(20,34,33,35)(24,39,38,45)(25,32,40,46)(28,42,44,43)(37,41,47,48)Polycyclic presentation:
2^9|a^2=h,c^2=g,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6657250);