Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456670 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1165 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)(5,6,10,15,9,16,11,13)(7,17,20,28,18,12,24,26)(8,21,23,14,22,29,25,30)(19,33,34,27,32,36,35,31);(5,7,10,24,9,18,11,20)(6,12,13,26,16,17,15,28)(8,19,23,35,22,32,25,34)(14,27,21,36,30,31,29,33);(2,4)(6,13)(7,18)(8,22)(10,11)(12,26)(14,21)(15,16)(17,28)(19,32)(27,33)(29,30)(31,36);(5,8)(6,14)(7,19)(9,22)(10,23)(11,25)(12,27)(13,21)(15,29)(16,30)(17,31)(18,32)(20,34)(24,35)(26,36)(28,33)Polycyclic presentation:
2^9|a^2=f,b^2=egh,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,f^c=fi,h^c=hi,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, 379090);