Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,14,11,16,9,15)(7,12,19,27,20,28,18,32)(8,21,24,13,23,31,25,30)(17,29,34,26,33,36,22,35);(1,2,3,4)(5,7)(6,12)(8,22)(9,18)(10,19)(11,20)(13,29)(14,27)(15,32)(16,28)(17,24)(21,35)(23,34)(25,33)(26,31)(30,36);(5,8)(6,13)(7,17)(9,25)(10,24)(11,23)(12,26)(14,31)(15,21)(16,30)(18,22)(19,34)(20,33)(27,36)(28,35)(29,32);(8,23)(13,30)(17,33)(21,31)(22,34)(24,25)(26,35)(29,36)Polycyclic presentation:
2^9|a^2=h,b^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,g^b=gi,d^c=di,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72256485, #72256488, #72256489, … |
| maximal normal subgroup of | #72256485, #72256488, #72256489, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264919);