Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196152 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1700 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)(3,7)(4,6)(5,8)(9,10)(11,17,20,24)(12,15)(13,16)(14,19,18,23)(21,22);(1,3,5,4)(2,6,8,7)(9,10)(11,18,20,14)(12,21)(13,16)(15,22)(17,23,24,19);(1,3,5,4)(2,7,8,6)(9,11,12,19,13,20,22,23)(10,14,21,24,16,18,15,17);(1,4,5,3)(2,7,8,6)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=efg,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72227237, #72227274, #72227287, … |
| maximal normal subgroup of | #72227237, #72227274, #72227287, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276558);