Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233184 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #121 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,17)(14,19)(15,20)(16,18)(21,22,27,31)(23,29,33,36)(24,34,32,28)(25,30,26,35);(5,7,9,16)(6,10,12,18)(8,14,17,15)(11,19,13,20)(21,23,27,33)(22,28,31,34)(24,25,32,26)(29,30,36,35);(2,4)(6,11)(7,14)(8,17)(12,13)(15,16)(19,20)(21,24,26,23,27,32,25,33)(22,29,30,34,31,36,35,28);(6,12)(10,18)(11,13)(19,20)(21,25,27,26)(22,30,31,35)(23,32,33,24)(28,36,34,29)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=e,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fhi,g^c=ghi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 146438);