Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478457 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
993 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)(7,11,17,24)(8,18)(9,13)(10,14)(12,20)(15,22,31,25)(16,27,33,23)(19,26)(21,29)(28,34)(30,35,32,36);(5,7)(6,11,14,24)(8,15)(9,16)(10,17)(12,25,18,22)(13,27,29,23)(19,32)(20,31)(21,33)(26,36,28,35)(30,34);(5,8)(6,12,14,18)(7,15)(9,19)(10,20)(11,22,24,25)(13,28,29,26)(16,32)(17,31)(21,34)(23,35,27,36)(30,33);(2,4)(5,9,10,21)(6,13,14,29)(7,16,17,33)(8,19,20,34)(11,23,24,27)(12,26,18,28)(15,30,31,32)(22,35,25,36)Polycyclic presentation:
2^9|a^2=eg,b^2=g,c^2=g,d^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,h^b=hi,d^c=de,f^c=fi,h^c=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72285240, #72285347, #72285387, … |
| maximal normal subgroup of | #72285240, #72285347, #72285387, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386870);