Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6347305 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3932 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
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)(16,22)(18,24)(21,23);(1,2)(3,8)(4,6)(5,7)(9,11)(10,14)(12,21)(13,18)(15,23)(16,24)(17,20)(19,22);(3,7)(5,8);(10,15)(14,23)(16,20)(17,24);(2,6)(5,8)(11,18)(14,23)(17,24)(19,21)Polycyclic presentation:
2^9|f^2=h,b^a=bf,c^a=cg,d^a=dhi,f^a=fh,e^b=eh,f^b=fh,g^b=gi,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232536, #72232543, #72232556, … |
| maximal normal subgroup of | #72232536, #72232543, #72232556, … |
| maximal quotient of | #72258023 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254501);