Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 52 points:
(2,4)(5,6,9,13,10,7,11,15)(8,12,17,19,16,14,18,20)(21,22)(23,33,37,43)(24,38,41,45)(25,31)(26,30)(27,32)(28,36,44,35)(29,39,46,40)(34,48)(42,51)(47,49)(50,52);(1,2,3,4)(5,7,9,15,10,6,11,13)(8,14,17,20,16,12,18,19)(21,23,25,35,27,37,26,36)(22,28,31,43,32,44,30,33)(24,34,39,49,41,51,40,50)(29,42,38,52,46,48,45,47);(5,8)(6,12)(7,14)(9,17)(10,16)(11,18)(13,19)(15,20)(21,24,27,41)(22,29,32,46)(23,34,37,51)(25,39,26,40)(28,42,44,48)(30,45,31,38)(33,47,43,52)(35,49,36,50);(8,16)(12,14)(17,18)(19,20)(21,25,27,26)(22,30,32,31)(23,35,37,36)(24,39,41,40)(28,33,44,43)(29,45,46,38)(34,49,51,50)(42,47,48,52)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^b=eh,f^b=fh,g^b=gh,d^c=di,e^c=eh,f^c=fh,g^c=gh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 100855);