Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470231 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 | |
4438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,7)(5,8)(9,10,14,13)(11,15,18,22)(12,16,21,20)(17,24,19,23);(3,4)(6,7)(9,11)(10,15)(12,19)(13,22)(14,18)(16,24)(17,21)(20,23);(1,3,5,4)(2,6,8,7)(9,12,14,21)(10,16,13,20)(11,17,18,19)(15,23,22,24);(1,4,5,3)(2,7,8,6)(9,13,14,10)(11,15,18,22)(12,20,21,16)(17,23,19,24);(2,8)(6,7)(10,13)(15,22)(16,20)(23,24)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,g^b=gi,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72284205, #72284232, #72284296, … |
| maximal normal subgroup of | #72284205, #72284232, #72284296, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394856);