Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| 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 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5249 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,14,17)(11,12,15,22)(13,16,23,24)(18,20,19,21);(2,4)(9,11)(10,12)(13,18)(14,15)(16,20)(17,22)(19,23)(21,24);(6,8)(11,18)(12,20)(13,23)(15,19)(16,24)(21,22);(9,12)(10,15)(11,17)(13,20)(14,22)(16,19)(18,24)(21,23);(11,18)(12,20)(13,23)(15,19)(16,24)(21,22)Polycyclic presentation:
2^9|a^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250240, #72256930, #72256938, … |
| maximal normal subgroup of | #72250240, #72256930, #72256938, … |
| maximal quotient of | #72264150, #72265178, #72306558, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480688);