Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535655 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1384 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)(4,8)(5,6)(17,18,23,27)(19,28,30,42)(20,33,36,47)(21,26,22,37)(24,31,39,32)(25,34,40,35)(29,43,44,38)(41,45,48,46);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,32,23,30,21,31)(18,24,26,28,27,39,37,42)(20,29,35,46,36,44,34,45)(25,38,33,41,40,43,47,48);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,21,34,23,36,22,35)(18,25,37,47,27,40,26,33)(19,29,31,45,30,44,32,46)(24,38,42,48,39,43,28,41);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16)(19,30)(24,39)(28,42)(29,44)(31,32)(38,43)(41,48)(45,46);(11,15)(14,16)(17,21,23,22)(18,26,27,37)(19,31,30,32)(20,34,36,35)(24,28,39,42)(25,33,40,47)(29,45,44,46)(38,41,43,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=fi,d^2=fgi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6463746);