Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454977 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23439 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1459 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)(5,6,9,15,11,12,10,25)(7,16,19,26,20,27,17,13)(8,21,22,35,24,14,23,30)(18,31,33,29,34,36,32,28);(6,12)(7,17)(9,10)(13,27)(14,21)(16,26)(18,32)(19,20)(22,23)(28,36)(29,31)(33,34);(2,4)(5,7)(6,13)(8,18)(9,17)(10,19)(11,20)(12,26)(14,29)(15,27)(16,25)(21,28)(22,32)(23,33)(24,34)(30,31)(35,36);(5,8,11,24)(6,14)(7,18,20,34)(9,22,10,23)(12,21)(13,28)(15,30)(16,31)(17,32,19,33)(25,35)(26,29)(27,36)Polycyclic presentation:
2^9|a^2=f,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362174);