Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454676 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#454 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1805 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)(5,6,10,15)(7,16,19,27)(8,20)(9,12,25,14)(11,24,28,18)(13,22)(17,32)(21,30)(23,29)(26,35)(31,34)(33,36);(5,7,10,19)(6,11,15,28)(8,17,23,35)(9,24,25,18)(12,16,14,27)(13,26,30,32)(20,36,29,31)(21,34,22,33);(2,4)(6,12)(8,21)(9,25)(11,16)(13,30)(14,15)(17,33)(18,24)(22,23)(27,28)(31,36)(34,35);(5,8)(6,13)(7,17)(9,22)(10,23)(11,26)(12,29)(14,20)(15,30)(16,31)(18,34)(19,35)(21,25)(24,33)(27,36)(28,32)Polycyclic presentation:
2^9|a^2=ef,b^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361874);