Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483882 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33582 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
720 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)(7,15,17,33)(8,19)(9,22)(11,23,25,35)(13,28)(14,27)(16,32,18,20)(21,31)(24,29,30,36)(26,34);(1,2,3,4)(5,6,10,12)(7,11,17,25)(8,13,21,26)(9,14,22,27)(15,23,33,35)(16,30,18,24)(19,28,31,34)(20,29,32,36);(5,7)(6,11)(8,20)(9,16)(10,17)(12,25)(13,29)(14,30)(15,31)(18,22)(19,33)(21,32)(23,34)(24,27)(26,36)(28,35);(6,12)(7,16,17,18)(8,21)(11,24,25,30)(14,27)(15,32,33,20)(19,31)(23,29,35,36)Polycyclic presentation:
2^9|a^2=g,b^2=ei,d^2=g,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,d^c=dh,f^c=fhi,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250341, #72250872, #72265101, … |
| maximal normal subgroup of | #72250341, #72250872, #72265101, … |
| maximal quotient of | #72318674, #72318762, #72319819, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403037);