Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8738 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #381 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#381 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1978 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,20,23,15)(12,17)(13,19)(14,18)(16,22,24,21);(9,11)(10,15)(12,21)(13,22)(14,23)(16,19)(17,24)(18,20);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16,18,24)(12,21)(13,22)(14,23)(15,19,20,17);(9,12,14,13)(10,17,18,19)(11,21,23,22)(15,24,20,16)Polycyclic presentation:
2^9|a^2=g,c^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=gh,g^c=ghi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72255972, #72256011, #72256028, … |
| maximal normal subgroup of | #72255972, #72256011, #72256028, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 97238);