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 | |
3255 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)(11,15)(12,18,21,16)(13,19)(14,17)(20,23,22,24);(1,2,4,5)(3,6,7,8)(9,11,15,10)(12,16,21,18)(13,19,14,17)(20,23,22,24);(2,5)(3,7)(9,12)(10,16)(11,18)(13,22)(14,20)(15,21)(17,24)(19,23);(1,3)(2,6)(4,7)(5,8)(9,13)(10,17)(11,19)(12,20)(14,15)(16,23)(18,24)(21,22);(2,5)(3,7)(9,14,15,13)(10,17,11,19)(12,20,21,22)(16,24,18,23)Polycyclic presentation:
2^9|a^2=fi,b^2=h,e^2=i,b^a=bf,c^a=cg,d^a=di,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 | #72316108, #72317612, #72317623, … |
| maximal normal subgroup of | #72316108, #72317612, #72317623, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503391);