Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417562 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1981 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)(7,15,18,22)(11,24,16,28)(17,29,31,33)(25,35,30,36);(5,6)(7,16,18,11)(8,12)(9,21)(10,14)(13,23)(15,24,22,28)(17,25,31,30)(19,32)(20,27)(26,34)(29,36,33,35);(5,7)(6,11)(8,17)(9,22)(10,18)(12,25)(13,28)(14,16)(15,23)(19,33)(20,31)(21,24)(26,36)(27,30)(29,34)(32,35);(2,4)(5,8,10,20)(6,12,14,27)(7,17,18,31)(9,19,23,34)(11,25,16,30)(13,26,21,32)(15,29,22,33)(24,35,28,36)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72262980, #72263055, #72263056, … |
| maximal normal subgroup of | #72262980, #72263055, #72263056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325168);