Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4769 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,13,15)(11,17,19,12)(14,16,23,24)(18,21,20,22);(2,4)(9,11)(10,12)(13,19)(14,18)(15,17)(16,22)(20,23)(21,24);(6,8)(10,15)(11,18)(12,21)(14,23)(17,22)(19,20);(9,12)(10,11)(13,17)(14,22)(15,19)(16,18)(20,24)(21,23);(11,18)(12,22)(14,23)(16,24)(17,21)(19,20)Polycyclic presentation:
2^9|a^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250240, #72256979, #72256990, … |
| maximal normal subgroup of | #72250240, #72256979, #72256990, … |
| maximal quotient of | #72264150, #72265178, #72308781, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480654);