Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10564 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #394 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#394 |
| prime radical | #60734 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1290 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)(9,10)(11,12)(13,18)(14,19)(15,20)(16,17)(21,23)(22,24);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,18)(13,21)(14,17)(15,22)(19,24)(20,23);(5,8)(9,12,13,23,15,24,14,16)(10,17,19,22,20,21,18,11);(9,13,15,14)(10,18,20,19)(11,21,22,17)(12,23,24,16)Polycyclic presentation:
2^9|c^2=e,d^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^b=gh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72310573, #72310575, #72310577, … |
| maximal normal subgroup of | #72310573, #72310575, #72310577, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 146550);