Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418976 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3019 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2291 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,17,12,18,11,16)(7,19,21,13,22,31,28,32)(8,23,26,34,9,15,25,14)(20,35,24,30,36,29,27,33);(5,7,10,21,12,22,11,28)(6,13,16,19,18,32,17,31)(8,24,25,20,9,27,26,36)(14,33,23,35,34,30,15,29);(5,8,12,9)(6,14,18,34)(7,20,22,36)(10,25,11,26)(13,29,32,35)(15,16,23,17)(19,33,31,30)(21,27,28,24);(2,4)(5,9,12,8)(6,15,18,23)(7,20,22,36)(10,25,11,26)(13,30,32,33)(14,16,34,17)(19,35,31,29)(21,24,28,27)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,g^b=gi,f^c=fi,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, 326205);