Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1521 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)(17,18,23,26,22,28,21,27)(19,24,36,39,30,38,31,40)(20,32,33,25,35,42,34,41)(29,43,45,37,44,48,46,47);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,24,28,38)(20,29,35,44)(21,31,23,36)(25,37,41,47)(26,39,27,40)(32,43,42,48)(33,45,34,46);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20)(18,25)(19,29)(21,33)(22,35)(23,34)(24,37)(26,32)(27,42)(28,41)(30,44)(31,45)(36,46)(38,47)(39,43)(40,48);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(19,30)(24,38)(29,44)(31,36)(37,47)(39,40)(43,48)(45,46);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(17,21,22,23)(18,26,28,27)(19,31,30,36)(20,33,35,34)(24,39,38,40)(25,32,41,42)(29,45,44,46)(37,43,47,48)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=f,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6458982);