Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632088 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35660 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2275 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)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,29,32,24)(20,33,35,42)(21,28,23,26)(25,37,40,34)(30,44,46,48)(31,39,36,41)(38,47,43,45);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,32)(18,24,27,29)(20,30,35,46)(21,36,23,31)(25,38,40,43)(26,41,28,39)(33,48,42,44)(34,47,37,45);(11,15)(13,16)(17,20,22,35)(18,25,27,40)(19,30,32,46)(21,34,23,37)(24,38,29,43)(26,33,28,42)(31,45,36,47)(39,44,41,48);(2,6)(3,8)(11,15)(13,16)(17,19)(18,24)(20,30)(21,36)(22,32)(23,31)(25,38)(26,41)(27,29)(28,39)(33,48)(34,47)(35,46)(37,45)(40,43)(42,44);(2,6)(5,7)(10,14)(13,16)(17,21,22,23)(18,26,27,28)(19,31,32,36)(20,34,35,37)(24,39,29,41)(25,33,40,42)(30,45,46,47)(38,44,43,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6539303);