Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394283 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60359 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2673 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)(7,16)(8,20)(9,25)(10,14)(11,15)(12,18)(13,23)(17,27)(19,28)(21,30)(22,32)(24,31)(26,29)(33,35)(34,36);(5,7,9,18,11,19,10,26)(6,12,14,16,15,29,25,28)(8,21,22,17,24,33,23,34)(13,30,31,36,32,35,20,27);(5,8)(6,13)(7,17)(9,22)(10,23)(11,24)(12,27)(14,31)(15,32)(16,30)(18,33)(19,34)(20,25)(21,26)(28,35)(29,36);(2,4)Polycyclic presentation:
2^9|b^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72313084, #72313103, #72313151 |
| maximal normal subgroup of | #72313084, #72313103, #72313151 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301479);