Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18866 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#379 |
| prime radical | #62800 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2463 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5,6,7)(9,10)(11,19)(12,15)(13,17)(14,18)(16,20)(21,24)(22,23);(1,2)(3,7)(4,6)(5,8)(10,15)(12,13)(16,23)(17,18)(19,24)(20,21);(9,11)(10,16)(12,20)(13,21)(14,22)(15,23)(17,19)(18,24);(9,12,14,13)(10,17,18,15)(11,20,22,21)(16,19,24,23)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fh,g^b=ghi,e^c=eh,f^c=fh86 conjugacy classes — view the full character table.
| maximal subgroup of | #72239103 |
| maximal normal subgroup of | #72239103 |
| maximal quotient of | #72239136 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 93699);