Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | #29725 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1521 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)(9,10,12,13)(11,14,15,16)(17,18)(19,30)(20,35)(21,37)(22,28)(23,29)(24,27)(25,34)(26,36)(31,42)(32,43)(33,46)(38,44)(39,48)(40,47)(41,45);(1,3)(2,5)(4,8)(6,7)(17,19)(18,25)(20,22)(21,32)(23,34)(24,41)(26,28)(27,43)(29,30)(31,33)(35,48)(36,39)(37,45)(38,40)(42,44)(46,47);(10,13)(14,16)(17,20,24,33,23,36,21,38)(18,26,27,44,29,35,37,46)(19,31,32,22,34,40,41,39)(25,42,45,28,30,47,43,48);(2,6)(5,7)(17,21,23,24)(18,27,29,37)(19,32,34,41)(20,33,36,38)(22,31,39,40)(25,43,30,45)(26,46,35,44)(28,47,48,42);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(17,22,21,31,23,39,24,40)(18,28,37,42,29,48,27,47)(19,33,41,20,34,38,32,36)(25,44,43,26,30,46,45,35)Polycyclic presentation:
2^9|a^2=gi,c^2=h,d^2=i,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6739170);