Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518920 |
| 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 | #45412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1569 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,24,29)(19,25,32,42)(20,33,38,44)(21,39,31,41)(22,27,23,28)(26,36,45,37)(30,43,46,48)(34,40,35,47);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,24,32)(18,25,29,42)(20,34,38,35)(21,22,31,23)(26,43,45,48)(27,41,28,39)(30,36,46,37)(33,40,44,47);(10,14)(11,16)(17,20,23,37,24,38,22,36)(18,26,27,33,29,45,28,44)(19,30,31,34,32,46,21,35)(25,40,39,43,42,47,41,48);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,22,32,24,31,23,19)(18,25,28,41,29,42,27,39)(20,35,36,46,38,34,37,30)(26,40,44,48,45,47,33,43);(2,5)(6,8)(10,14)(13,15)(17,22,24,23)(18,27,29,28)(19,31,32,21)(20,36,38,37)(25,41,42,39)(26,33,45,44)(30,34,46,35)(40,48,47,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,d^2=fgi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7502536);