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 | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1383 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)(9,10,13,15)(11,14,12,16)(17,18)(19,30)(20,33)(21,39)(22,28)(23,29)(24,27)(25,32)(26,37)(31,41)(34,43)(35,44)(36,45)(38,48)(40,42)(46,47);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19)(18,25)(20,34)(21,22)(23,32)(24,40)(26,43)(27,28)(29,30)(31,38)(33,45)(35,47)(36,37)(39,42)(41,44)(46,48);(9,12,13,11)(10,14,15,16)(17,20,24,38,23,37,21,35)(18,26,27,44,29,33,39,48)(19,31,22,34,32,47,40,36)(25,41,42,43,30,46,28,45);(2,6)(5,7)(17,21,23,24)(18,27,29,39)(19,22,32,40)(20,35,37,38)(25,28,30,42)(26,44,33,48)(31,34,47,36)(41,45,46,43);(3,8)(5,7)(17,22,24,32,23,40,21,19)(18,28,27,25,29,42,39,30)(20,36,38,31,37,34,35,47)(26,43,44,46,33,45,48,41)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=ghi,d^2=i,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6727419);