Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #254467 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #32899 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1643 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)(3,5)(4,6)(7,8)(15,18)(17,24);(1,3)(2,5)(4,7)(6,8)(9,10,14,12)(11,17)(13,16,23,21)(15,22)(18,20)(19,24);(5,8)(9,11)(10,15)(12,18)(13,22)(14,19)(16,24)(17,21)(20,23);(9,12,14,10)(11,18,19,15)(13,21,23,16)(17,20,24,22)Polycyclic presentation:
2^9|b^2=eh,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fi,e^c=eh,g^c=gh,f^d=fh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72233417, #72255859, #72256011, … |
| maximal normal subgroup of | #72233417, #72255859, #72256011, … |
| maximal quotient of | #72233387, #72238485, #72262557, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 161670);