Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #267659 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,13,15,10)(8,16,17,11)(9,12)(14,18)(19,20)(21,22,24,27,25,28,33,36)(23,29,30,34,31,35,32,26);(1,2)(3,4)(5,7,9,15)(6,10,12,13)(8,14,17,20)(11,18,16,19);(2,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,18)(16,17)(19,20)(21,23,25,31)(22,26,28,34)(24,32,33,30)(27,35,36,29);(5,8)(6,11)(7,14)(9,17)(10,18)(12,16)(13,19)(15,20)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,e^c=eh,f^c=fi,f^d=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 174889);