Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6354139 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1931 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)(3,5)(4,6)(7,8)(10,15)(11,19)(12,13)(17,18)(20,21)(22,23);(9,10,12,17,14,18,13,15)(11,16,19,23,20,24,21,22);(2,5)(3,8)(6,7)(9,11)(10,16)(12,19)(13,21)(14,20)(15,22)(17,23)(18,24);(2,6)(5,7)(9,12,14,13)(10,17,18,15)(11,19,20,21)(16,23,24,22);(11,20)(16,24)(19,21)(22,23)Polycyclic presentation:
2^9|b^2=fh,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,e^c=eh,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72306659, #72306674, #72306680, … |
| maximal normal subgroup of | #72306659, #72306674, #72306680, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261339);