Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324564 |
| 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 | #53065 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18,22,23)(19,27,30,44)(20,25,31,38)(21,35,36,26)(24,39,37,29)(28,45,32,48)(33,43,34,42)(40,46,41,47);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(18,23)(20,31)(24,37)(26,35)(27,44)(28,32)(33,34)(46,47);(2,6)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,21,29,22,30,36,39)(18,24,26,27,23,37,35,44)(20,32,34,46,31,28,33,47)(25,41,43,48,38,40,42,45);(3,8)(5,7)(10,14)(11,16)(17,20,22,31)(18,25,23,38)(19,28,30,32)(21,34,36,33)(24,40,37,41)(26,43,35,42)(27,45,44,48)(29,47,39,46);(2,6)(5,7)(10,14)(13,15)(17,21,22,36)(18,26,23,35)(19,29,30,39)(20,33,31,34)(24,27,37,44)(25,42,38,43)(28,46,32,47)(40,48,41,45)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=fgi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dhi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7234495);