Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418863 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1891 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,28,12,16,11,17)(7,18,20,30,9,13,21,15)(8,22,27,32,25,33,26,14)(19,35,23,29,24,31,36,34);(1,2)(3,4)(5,7,12,9)(6,13,16,18)(8,23,25,36)(10,21,11,20)(14,31,32,35)(15,28,30,17)(19,26,24,27)(22,29,33,34);(2,4)(5,8)(6,14)(7,19)(9,24)(10,27)(11,26)(12,25)(13,29)(15,31)(16,32)(17,33)(18,34)(20,23)(21,36)(22,28)(30,35);(5,9,12,7)(6,15,16,30)(8,24,25,19)(10,20,11,21)(13,17,18,28)(14,31,32,35)(22,29,33,34)(23,26,36,27)Polycyclic presentation:
2^9|a^2=ef,b^2=i,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326062);