Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573715 |
| 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 | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,14)(12,17,20,15)(13,18)(16,22)(19,23,21,24);(1,2,4,5)(3,6,7,8)(9,11,14,10)(12,15,20,17)(13,18,22,16)(19,23,21,24);(2,5)(3,7)(9,12)(10,15)(11,17)(13,21)(14,20)(16,24)(18,23)(19,22);(1,3)(2,6)(4,7)(5,8)(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24);(2,5)(3,7)(10,11)(13,22)(15,17)(19,21)Polycyclic presentation:
2^9|a^2=fi,b^2=h,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72317595, #72317630, #72317652, … |
| maximal normal subgroup of | #72317595, #72317630, #72317652, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503397);