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 | |
3483 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)(12,16)(13,14)(17,24)(18,19)(20,23)(21,22);(1,2,4,5)(3,6,7,8)(9,11,14,19)(10,15,13,18)(12,20,21,17)(16,24,22,23);(2,5)(3,7)(9,12)(10,16)(11,17)(13,22)(14,21)(15,23)(18,24)(19,20);(1,3)(2,6)(4,7)(5,8)(9,13,14,10)(11,18,19,15)(12,16,21,22)(17,23,20,24);(2,5)(3,7)(10,13)(11,19)(16,22)(17,20)Polycyclic presentation:
2^9|b^2=h,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,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 | #72317614, #72317620, #72317652, … |
| maximal normal subgroup of | #72317614, #72317620, #72317652, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6502983);