Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1411 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)(11,14)(12,13)(15,16)(17,18,24,30)(19,25,32,39)(20,26,36,43)(21,27,23,29)(22,28,37,44)(31,38,45,47)(33,40,46,48)(34,41,35,42);(1,3,4,8)(2,5,6,7)(17,19)(18,25)(20,33)(21,22)(23,37)(24,32)(26,40)(27,28)(29,44)(30,39)(31,35)(34,45)(36,46)(38,42)(41,47)(43,48);(10,13)(14,16)(17,20,21,34,24,36,23,35)(18,26,27,41,30,43,29,42)(19,31,37,46,32,45,22,33)(25,38,44,48,39,47,28,40);(2,6)(5,7)(17,21,24,23)(18,27,30,29)(19,22,32,37)(20,34,36,35)(25,28,39,44)(26,41,43,42)(31,33,45,46)(38,40,47,48);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(17,22,21,19,24,37,23,32)(18,28,27,25,30,44,29,39)(20,33,34,31,36,46,35,45)(26,40,41,38,43,48,42,47)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=hi,d^2=i,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6737190);