Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476609 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
659 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)(7,13,19,18)(8,22)(9,11)(10,16)(12,15)(14,25)(17,31,34,36)(20,28,21,30)(23,29,33,35)(24,27)(26,32);(5,7,12,21,10,19,11,20)(6,13,15,30,16,18,9,28)(8,23,25,34,27,33,26,17)(14,31,24,29,32,36,22,35);(5,8,12,25,10,27,11,26)(6,14,15,24,16,32,9,22)(7,17,21,23,19,34,20,33)(13,29,30,36,18,35,28,31);(2,4)(5,9,12,16,10,15,11,6)(7,18,21,30,19,13,20,28)(8,24,26,32,27,22,25,14)(17,29,33,36,34,35,23,31)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,c^2=ef,d^2=fi,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,h^a=hi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386749);