Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#379 |
| prime radical | C22×H(5,𝔽2) | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
3513 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,22)(19,24)(21,23);(9,11)(10,14)(12,21)(13,19)(15,23)(16,24)(17,20)(18,22);(12,22)(16,20)(17,24)(18,21);(2,5)(6,8)(10,15)(14,23)(16,20)(17,24)Polycyclic presentation:
2^9|a^2=g,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,e^b=eh,f^b=fh,e^c=eh98 conjugacy classes — view the full character table.
| maximal subgroup of | #72239485, #72278516, #72278605, … |
| maximal normal subgroup of | #72239485, #72278516, #72278605, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92138);