Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421163 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| 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 | |
793 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,13,20,31)(8,23)(9,25)(10,17)(11,16)(12,15)(14,28)(18,33,24,36)(19,34,29,35)(21,32,22,30)(26,27);(5,7,10,21,12,20,11,22)(6,13,16,30,15,31,17,32)(8,24,25,35,14,18,26,34)(9,19,23,36,27,29,28,33);(5,8,12,14)(6,9,15,27)(7,18,20,24)(10,26,11,25)(13,29,31,19)(16,28,17,23)(21,35,22,34)(30,36,32,33);(2,4)(5,9)(6,14)(7,19,20,29)(8,15)(10,28)(11,23)(12,27)(13,24,31,18)(16,25)(17,26)(21,33,22,36)(30,34,32,35)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356273);