Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357777 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,17)(12,20)(13,15)(14,19)(16,21)(18,23)(22,24);(9,11,12,19,13,18,21,24)(10,14,16,17,15,22,20,23);(1,2,3,5,4,6,7,8)(10,15)(14,22)(16,20)(17,23);(2,6)(5,8);(11,18)(14,22)(17,23)(19,24)Polycyclic presentation:
2^9|b^2=f,c^2=gh,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,e^b=eh,d^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72214347, #72283748, #72306698, … |
| maximal normal subgroup of | #72214347, #72283748, #72306698, … |
| maximal quotient of | #72224781 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264975);