Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417690 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1597 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)(7,19,9,15,22,34,21,13)(8,23,27,33,28,32,26,14)(20,29,24,30,36,31,25,35);(5,7)(6,13)(8,24)(9,10)(11,21)(12,22)(14,30)(15,18)(16,34)(17,19)(20,26)(23,31)(25,28)(27,36)(29,32)(33,35);(2,4)(5,8,12,28)(6,14,18,33)(7,20,22,36)(9,25,21,24)(10,26,11,27)(13,29,15,31)(16,23,17,32)(19,35,34,30);(5,9,11,22,12,21,10,7)(6,15,17,19,18,13,16,34)(8,25,27,36,28,24,26,20)(14,31,32,35,33,29,23,30)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327122);