Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996671 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2121 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(18,21)(19,23)(22,24);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(18,22)(21,24);(1,4,5,2)(3,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,18,23,24,20,22,19,21);(3,8)(6,7)(11,16)(13,15);(10,14)(13,15)(17,19,20,23)(18,21,22,24)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fgi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72256922, #72256931, #72256987, … |
| maximal normal subgroup of | #72256922, #72256931, #72256987, … |
| maximal quotient of | #72265490 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6927912);