Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1340 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)(17,18,23,27)(19,28,32,45)(20,33,35,43)(21,26,22,38)(24,30,40,31)(25,37,42,36)(29,41,34,39)(44,47,48,46);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,30,23,32,22,31)(18,24,38,45,27,40,26,28)(20,34,37,46,35,29,36,47)(25,41,43,44,42,39,33,48);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20,22,36,23,35,21,37)(18,25,26,33,27,42,38,43)(19,29,31,46,32,34,30,47)(24,39,28,44,40,41,45,48);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(20,35)(25,42)(29,34)(33,43)(36,37)(39,41)(44,48)(46,47);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(17,21,23,22)(18,26,27,38)(19,30,32,31)(20,36,35,37)(24,28,40,45)(25,43,42,33)(29,46,34,47)(39,48,41,44)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,d^2=fgi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^c=di,e^c=ei,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6447996);