Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433057 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1710 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)(11,13)(12,14)(15,16)(17,18)(19,29)(20,34)(21,26)(22,39)(23,28)(24,32)(25,37)(27,40)(30,43)(31,42)(33,45)(35,41)(36,47)(38,48)(44,46);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,32,23,33,40,35)(18,24,27,29,28,41,39,45)(20,30,37,46,36,21,31,38)(25,26,34,44,42,43,47,48);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,20,22,37,23,36,40,31)(18,25,27,34,28,42,39,47)(19,30,32,46,33,21,35,38)(24,26,29,44,41,43,45,48);(1,4,5,3)(2,6,8,7)(10,14)(13,16)(17,21,23,30)(18,26,28,43)(19,31,33,37)(20,35,36,32)(22,38,40,46)(24,34,41,47)(25,29,42,45)(27,44,39,48);(20,36)(21,30)(25,42)(26,43)(31,37)(34,47)(38,46)(44,48)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341510);