Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456510 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,18)(8,21)(9,25,29,32)(10,16)(11,15)(12,17)(13,20)(14,28,33,27)(19,23,35,30)(22,24)(26,31,36,34);(1,2)(3,4)(5,7,10,20,12,22,11,21)(6,13,15,18,17,8,16,24)(9,19,28,34,29,35,27,31)(14,30,32,26,33,23,25,36);(2,4)(5,8,12,13)(6,7,17,22)(9,26,29,36)(10,18,11,24)(14,31,33,34)(15,21,16,20)(19,32,35,25)(23,28,30,27);(5,9)(6,14)(7,19)(8,23)(10,28)(11,27)(12,29)(13,30)(15,32)(16,25)(17,33)(18,26)(20,34)(21,31)(22,35)(24,36)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,c^2=i,e^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,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363709);