Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499054 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#481 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
596 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)(5,6,9,15,12,7,10,14)(8,19,22,17,24,31,21,13)(11,18,25,32,29,16,26,27)(20,23,33,36,30,34,35,28);(5,7,12,6)(8,20)(9,15,10,14)(11,27,29,32)(13,23)(16,26,18,25)(17,34)(19,28)(21,33)(22,35)(24,30)(31,36);(2,4)(5,8,10,22,12,24,9,21)(6,13,14,19,7,17,15,31)(11,28,26,23,29,36,25,34)(16,33,32,30,18,35,27,20);(5,9,12,10)(6,14,7,15)(8,21,24,22)(11,25,29,26)(13,19,17,31)(16,32,18,27)(20,33,30,35)(23,28,34,36)Polycyclic presentation:
2^9|a^2=f,b^2=eh,c^2=e,d^2=i,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72322580, #72322593, #72322607, … |
| maximal normal subgroup of | #72322580, #72322593, #72322607, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406620);