Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213798 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35527 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2072 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)(3,5)(4,6)(7,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,24)(21,23);(1,3)(2,5)(4,7)(6,8)(9,11,13,19,14,20,12,21)(10,15,16,24,17,18,22,23);(2,6)(5,8)(11,19)(12,13)(15,23)(16,22)(18,24)(20,21);(3,7)(5,8)(11,20)(12,13)(15,18)(16,22)Polycyclic presentation:
2^9|b^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,h^a=hi,c^b=cg,d^b=def,e^b=ei,d^c=de,g^c=gi,h^c=hi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72235221, #72235228, #72235229, … |
| maximal normal subgroup of | #72235221, #72235228, #72235229, … |
| maximal quotient of | #72235262, #72235370 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298916);