Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324500 |
| 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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1397 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(9,10)(11,14)(12,15)(13,16)(17,18,23,27)(19,24,22,36)(20,30,33,38)(21,26,35,42)(25,28,40,31)(29,37,34,41)(32,44,46,47)(39,43,48,45);(1,2)(3,7)(4,6)(5,8)(9,11)(10,13)(12,14)(15,16)(17,19,23,22)(18,24,27,36)(20,31,33,28)(21,34,35,29)(25,38,40,30)(26,41,42,37)(32,43,46,45)(39,44,48,47);(1,3)(2,5)(4,8)(6,7)(9,12,15,10)(11,13,16,14)(17,20,19,28,23,33,22,31)(18,25,36,38,27,40,24,30)(21,32,29,43,35,46,34,45)(26,39,41,47,42,48,37,44);(2,6)(3,8)(9,13,15,14)(10,16,12,11)(17,21,23,35)(18,26,27,42)(19,29,22,34)(20,32,33,46)(24,37,36,41)(25,39,40,48)(28,43,31,45)(30,44,38,47);(2,6)(5,7)(9,14,15,13)(10,16,12,11)(20,33)(21,35)(25,40)(26,42)(28,31)(29,34)(30,38)(37,41)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dhi,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7243406);